A20004056 中南大学 唐高朋 田家凯 余道顺 - 图文

2010高教社杯全国大学生数学建模竞赛

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日期： 2010 年 9 月 13 日

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2010高教社杯全国大学生数学建模竞赛

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A题 储油罐的变位识别与罐容表标定

摘要

本文先建立平放小椭圆型储油罐油位高度与对应的油量的函数解析式，然后利用给出的数据进行了误差检验，对模型进行修正，逐步深入，最后建立针对实际储油罐罐容表标定的模型及识别储油罐的变位的方法。

为研究罐体变位后对罐容表的影响，先建立倾角?=0°即罐体无变位情况下的罐容表模型，以小椭圆型储油罐为例，首先建立平放小椭圆型储油罐罐内油位高度与储油量的函数模型。然后用实际数据进行检验时，平均误差为3.37%，因此对模型进行适当修正，得到小椭圆型储油罐平放时候罐容表实际模型。 当小椭圆型储油罐发生纵向倾斜时，根据油位高度h的大小分为三种情况讨论:①油面超过油位探针最低点但并未完全覆盖油罐底面时，②油面完全浸过下底面，但是油面还未到达油罐上顶面最低点时，③油面到达油浮子上顶面却未浸没油位探针的最高点时.利用定积分求出垂直纵轴的平面面积的函数表达式，再利用对纵轴积分建立了各情况下的函数模型。然后对模型进行检验，平均误差为2.72%，因此对模型进行修正，对修正后的模型进行检验，得平均误差为0.67%，故修正后的模型可靠。最终得到罐体变位后对罐容表模型即罐体变位后对罐容表的影响，然后利用此模型在matlab7.0平台上编程给出罐体变位后油位高度间隔为1㎝的罐容表标定值。

在建立实际储油罐变位时定罐容表的数学模型时，跟据油位高度h的大小分为三种情况讨论：①当油面还未到达右边的球冠体部分,②当油面均到达了左右两边的球冠体,③当油面已经充满了左边的球冠体。同时将实际储油罐分为两部分：圆柱体和两端的球冠体。通过平行截面面积为已知的立体和平底曲面立体体积的求法，分别列出两个球冠体及圆柱体油面下的体积函数式，相加后利用Matlab7.0符号积分得变位后储油量V关于油面高度h的函数，函数含有参变量纵向倾斜角度?和横向偏转角度?，基于最小二乘原理，利用实际测得的油面高度和对应的出油量的数据，求出参数值?=2.12°、?=4.38°。将参数值代入模型给出罐体变位后油位高度间隔为10cm的罐容表标定值。并根据收集的显示油量容积求出残差为7.7332?10?5，可知模型贴切实际，可靠性高。作出随着参数?、?数值的变化储油量V关于油面高度h的函数图形，通过其走势识别油罐变位程度。

关键词：最小二乘原理 残差检验 定积分 储油罐变位

1

一、问题重述

储油罐采用流量计和油位计来测量进/出油量与罐内油位高度等数据，通过预先标定的罐容表（即罐内油位高度与储油量的对应关系）进行实时计算，以得到罐内油位高度和储油量的变化情况。

罐体的位置发生纵向倾斜和横向偏转等变化时，从而导致罐容表发生改变。因此，需要定期对罐容表进行重新标定。典型的储油罐尺寸及形状是其主体为圆柱体，两端为球冠体。

用数学建模方法研究解决储油罐的变位识别与罐容表标定的问题。 （1）为了掌握罐体变位后对罐容表的影响，利用小椭圆型储油罐（两端平头的椭圆柱体），分别对罐体无变位和倾斜角为?=4.1°的纵向变位两种情况做了实验，得到的实验数是油位高和油容量对应的表格。建立数学模型研究罐体变位后对罐容表的影响，并给出罐体变位后油位高度间隔为1cm的罐容表标定值。

（2）对于实际储油罐，试建立罐体变位后标定罐容表的数学模型，即罐内储油量与油位高度及变位参数之间的一般关系。利用罐体变位后在进/出油过程中的实际检测数据，根据建立的数学模型确定变位参数，并给出罐体变位后油位高度间隔为10cm的罐容表标定值。并对模型进一步分析。

二、问题分析

问题中给定的油罐是规则的，本文考虑用积分的方法求出油罐罐内高度与储油量的对应关系。

要研究罐体变位后对罐容表（即罐内油位高度与储油量的对应关系）的影响，先研究罐体无变位的罐容表。建立根据油罐空间几何形状积分建立小椭圆型储油罐平放的时候罐内油位高度与储油量的对应关系的理想模型。当油罐发生纵向倾斜时，在无变位的基础上，利用二重积分的方法建立各情况下的模型得到罐体变位后对罐容表模型（罐体变位后对罐容表的影响）。

在建立实际储油罐变位定罐容表的数学模型时，文中将油罐分为两部分：圆柱体和两端的球冠体。根据几何关系，分别求出被油覆盖的两个球冠体及圆柱体的体积，圆柱体积和球冠体积加和得到含参变量纵向倾斜角度?和横向偏转角度?的油罐体积V关与油面不同高度h时的函数，基于最小二乘原理求出参数值。

三、模型假设

（1）储油罐不发生局部形变，是规则的； （2）油罐的围测值科学精确；

（3）储油罐周围的温度变化不影响各实验数据的收集； （4）收集数据的时间不影响实验数据的收集； （5）液体压力对油罐的形状不产生影响； （6）实验所收集得到的数据是准确可靠的。

四、符号说明

h：油罐中油位探针显示的油位高度；

h0：实际圆柱型两端为球冠体油罐横向和纵向倾斜的情况下油位探针截面处的

2

实际油面高度；

s1：椭圆油罐未倾斜时截面液体浸没部分的面积；

L1：两端平头的椭圆柱体的总长度；

?：油罐纵向倾斜的角度；

?：实际圆柱型油罐横向倾斜角度； r：油罐圆柱体横截面半径；

R：实际圆柱型油罐两端球冠的半径。

s2：圆柱型油罐横截面液体浸没部分的面积；

V1：未倾斜时油罐中的储油量； V2:纵向倾斜时小椭圆油罐中的储油量；

V3：实际圆柱型油罐在变位后圆柱部分的储油量；

V4:变位后圆柱形油罐总储油量。

五、模型的建立与求解

5.1问题一

5.1.1模型的建立 1、罐体无变位

为了求解小椭圆型储油罐在纵向倾斜后的储油量和油位高度间的关系，首先应先计算出小椭圆型储油罐未发生倾斜时罐内油位高度与储油量的对应的函数关系。根据椭圆油罐截面示意图可以建立直角坐标系，如下图所示：

x b a y 图1 椭圆油罐截面及其油面示意图

图中阴影部分为油浸没的截面，则可求出液体浸没部分的面积

h?b2a1?x2/b2dx （5.1.1） s1??? b通过使用微积分中的换元积分法可以得到如下关系式

s1?abarcsin(h?b?abab?h?b?)??sin?2arcsin()? （5.1.2） b22b??由于椭圆型储油罐并未倾斜纵向上面给横截面面积相等，则储油罐的油量为横截面积s1乘以椭圆柱体的总长度L1如下

3

h

图5 总模型函数图形

利用式（5.1.16）编写程序（见附录4），通过Matlab7.0软件求解得到罐体变位后油位高度间隔为1cm的罐容表标定值，见下表：

表6 罐体变位后油位高度间隔为1cm的罐容表

油位高度/m 储油量/m 油位高度/m 储油量/m 油位高度/m 储油量/m 油位高度/m 储油量/m

3

3

3

3

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15

0.0017 0.0035 0.0063 0.0100 0.0148 0.0207 0.0279 0.0363 0.0461 0.0574 0.0701 0.0844 0.1003 0.1177 0.1369 0.1578 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.6507 0.6816 0.7124 0.7432 0.7740 0.8048 0.8356 0.8664 0.8972 0.9280 0.9589 0.9939 1.0314 1.0692 1.1075 1.1461

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0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 1.8004 1.8429 1.8855 1.9283 1.9711 2.0140 2.0570 2.1000 2.1430 2.1861 2.2293 2.2724 2.3155 2.3586 2.4016 2.4446 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 3.1155 3.1556 3.1953 3.2348 3.2739 3.3126 3.3510 3.3889 3.4264 3.4635 3.5000 3.5300 3.5628 3.5956 3.6284 3.6611

0.16 0.1886 0.47 1.1850 0.78 2.4876 1.09 3.6939 0.17 0.2194 0.48 1.2243 0.79 2.5304 1.10 3.7267 0.18 0.2502 0.49 1.2639 0.80 2.5732 1.11 3.7595 0.19 0.281 0.5 1.3037 0.81 2.6159 1.12 3.7923 0.20 0.3118 0.51 1.3439 0.82 2.6585 1.13 3.8251 0.21 0.3426 0.52 1.3843 0.83 2.7010 1.14 3.8578 0.22 0.3734 0.53 1.4250 0.84 2.7433 1.15 3.8906 0.23 0.4043 0.54 1.4659 0.85 2.7855 1.16 3.9234 0.24 0.4351 0.55 1.5070 0.86 2.8275 1.17 3.9562 0.25 0.4659 0.56 1.5484 0.87 2.8693 1.18 3.9767 0.26 0.4967 0.57 1.5900 0.88 2.9109 1.19 3.9955 0.27 0.5275 0.58 1.6317 0.89 2.9523 1.20 4.0127 0.28 0.5583 0.59 1.6737 0.9 2.9935 0.29 0.5891 0.60 1.7158 0.91 3.0344 0.30 0.6199 0.61 1.7580 0.92 3.0751 5.2 问题二

5.2.1模型的建立

实际圆柱型储油罐在使用一段时间后，由于地基变形等原因，使罐体的位置会发生纵向倾斜和横向偏转等变化，从而导致罐容表发生改变，下面建立模型求解变位后圆柱型储油罐的储油量和倾斜角、探针显示油位等的关系式。

由于实际的圆柱型储油罐基本由圆柱体和两端的球冠体两部分构成，所以下面分两部分分别求解。

1、首先求解倾斜时圆柱部分的储油量，根据圆柱体无偏转倾斜的正截面图

可以建立平面直角坐标系如下图所示：

x r y h00 图10 圆柱型油罐截面及其油面示意图

图中阴影部分为油浸没的截面，则可求出液体浸没部分的面积：

s2???r?r?h02r2?x2dx （5.2.1）

其中，r为油罐两端球冠体和圆柱截面半径；h0为实际圆柱型两端为球冠体油罐横向和纵向倾斜的情况下油位探针截面处的实际油面高度。 然后，当圆柱型储油罐在发生纵向倾斜时，其储油量和实际油位高度间的关系并不是一个简单的公式就能求出的，此时液体的体积并不是一个规则的图形，即油面到圆柱油罐底部高度随圆柱纵轴线长度变化。根据圆柱储油罐正面示意图

10

的圆柱体部分建立平面直角坐标系，如下图所示：

t 油面 ? l

图11 圆柱储油罐正面示意图的圆柱体部分坐标系图

则由坐标系中的倾角?和纵轴的截距h0可以解得油面直线的方程为 t??ltan??h0 （5.2.2） 针截面处的实际油面高度；?为油罐纵向倾斜的角度；t为油面到圆柱储油罐圆柱体部分底部高度随椭圆柱长度l变化量。

下面利用二重积分的方法求解油罐圆柱部分的储油量时，根据实际油面高度（1）当油面浸没过油位探针的最低点但并未完全覆盖油罐圆柱部分的底面时，其示意图如下：

其中，h0为实际圆柱型两端为球冠体油罐横向和纵向倾斜的情况下油位探

d1 h0的大小分为三种情况讨论。

h0 d2 t ? 油面 l d2 d1

图12 油面高于探针最低点且未完全覆盖底面示意图

即：当0?h0?d2tan?时，利用图1的坐标系并结合式（5.2.1）和（5.2.2）使截面积对油罐轴向变量l积分可得变位后椭圆油罐中圆柱体部分的储油量：

h0?r?ltan??h0 V3??tan?dl??r2r2?x2dx （5.2.3）

?d1（2）当油面完全浸过油罐圆柱部分下底面，但是油面还未到达油罐圆柱部分上

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顶面时，其示意图如下：

t ? 油面 l d2 d1

图13 油面浸过油罐下底面且未到达上顶面示意图

即：当d2tan?﹤h0?2r-d1tan?时，利用图1的坐标系并结合式（5.2.1）和（5.2.2）使截面积对油罐轴向变量l积分可得纵向变位后实际油罐中圆柱部分的储油量：

?r?ltan??h0d V3??2dl??r 2r2?x2dx （5.2.3）

?d1（3）当油面到达油罐圆柱部分上顶面却未浸没油位探针的最高点时，其示意图如下所示：

t ? 油面 l d2

图14 油面到达上顶面且未浸没探针最高点示意图

即：当2r-d1tan??h0?2r时，利用图1的坐标系并结合式（5.2.1）和（5.2.2）使截面积对油罐轴向变量l积分可得变为后油罐中圆柱部分的储油量：

h?2r?r?ltan??h0d V3??r2(0)??2dl??r2r2?x2dx （5.2.4）

h0?2rtan?tan?其中，h0实际圆柱型两端为球冠体油罐横向和纵向倾斜的情况下油位探针截面处的实际油面高度；r为油罐两端球冠体和圆柱截面半径；?为油罐纵向倾斜的角度。

12

d1

2、实际的油罐的两端并不是平的，而是有两个球冠体构成，当油罐发生横向和纵向倾斜时球冠体中液体的体积并不是规则的，即没有特定的公式可以计算，所以下面利用曲面积分方法建立模型来求解左右两个球冠体中的液体容量。

（1）先考虑左边球冠体在倾斜时其中液体的容积，建立空间直角坐标系如下图所示：

x A B 液面 C h1 z D

d3 ?

图15 油罐倾斜情况下左球冠体坐标系示意图

其中，坐标原点是球冠体和圆柱体相交截面圆圆心，z轴为圆柱体轴向方向。 则根据球冠体的几何关系可以求解出球冠体的曲面方程为

z?R2?x2?y2?(R?d3) （5.2.5）

由图可得，该坐标系中油罐里液面的法向量为：??cos?,0,sin??，并且由图中可以看出液面经过点?h1?r,0,0?，则可以求出液面在x?o?z平面的直线方程为

???x?h1?r?cos??zsin??0 （5.2.6） ?y?0?联立球冠、液面方程，得方程 整理有

csc2?x2?2cot??R?d3?rcot??h1cot??x??R?d3?rcot??h1cot??2?R2?0根据韦达定理，

22x2??x?h?rcot??R?d???13??R ??b?b2?4ac交点横坐标值x1? 2a其中，a?csc2?；b?2cot??R?d3?rcot??h1cot??；

c??R?d3?rcot??h1cot??2?R2

由图5可以得液面浸没下球冠体体积分为两个部分，即不规则体ABC和BCD

由曲面积分的方法可以求得不规则体BCD的体积为：

13

V???[R2?x2?y2?(R?d3)]dxdy (5.2.7) BCDD1不规则体BCD的体积为：

??VABC????R2?x2?y2?(R?d3)??x?h1?r?cot??dxdy (5.2.8)

?D2?其中D1、D2为曲面积分的积分区域，其示意图如下所示：

图16 D1、D2积分区域示意图

则通过微积分的知识把曲面积分化成双重积分可以求解得：

2?x2h1?rrVBCD?2??rdx?0[R2?x2?y2?(R?d3)]dy (5.2.9)

22x VABC?2?1dx?r?x[R2?x2?y2?(R?d3)??x?h1?r?cot?]dy (5.2.10)

h1?r0即可得出左半部球冠体的储油量为

V左?VBCD+VABC (5.2.11)

其中，VBCD为图中不规则题BCD的体积，VABC为图中不规则体ABC的体积。 （2）计算左边球冠体在倾斜时其中液体的容积，同理，建立空间直角坐标系如下图所示：

14

z 液面 H F E

h2 G x ?

图17 油罐倾斜情况下右球冠体坐标系示意图

其中，坐标原点是球冠体和圆柱体相交截面圆圆心，z轴为圆柱体轴向方向。右边球冠体的曲面方程同左边球冠体，但是在右边球冠体中液面在x?o?z平面的直线方程为：

???x?h2?r?cos??zsin??0 （5.2.12） ?y?0?联立球冠、液面方程，得方程

22x2??x?h?rcot??(R?d)???23???R

整理有

csc2?x2?2cot??R?d3?rcot??h2cot??x??R?d3?rcot??h2cot??2?R2?0

?b?b2?4ac根据韦达定理，交点横坐标值 x2? 2a其中，a?csc2?；

b?2cot??R?d3?rcot??h2cot??； c??R?d3?rcot??h2cot??2?R2；

由图7可以得液面浸没下右球冠体体积为：

V右?VGHE? VEFH （5.2.13）

其中，VGHE为图中不规则体GHE的体积，VEFH为图中不规则体EFH的体积。 同理，计算不规则体的体积的时候仍可以使用曲面积分的方法，其积分区域

的示意图如下所示：

15

y H? x1 F? G F h1 H x

图18 右球冠体体积积分区域示意图

则可以计算出不规则体GHE的体积为：

22VGHE=2?rdx?0r?x[R2?x2?y2?(R?d3)]dy （5.2.14）

r?h2不规则体EFH的体积为：

r?h2r2?x22?x2?y2?(R?d)?(x?h?r)cot?]dy （5.2.15）VEFH=2?[R ?320x23、圆柱型储油罐基本由圆柱体和两端的球冠体两部分构成，所以变位后圆柱体

的总的储油量也应该由这两部分的储油量构成。

由于在求解变位后圆柱体部分的储油量的过程中根据液面到油罐底部的高度不同进行了分段计算，所以在计算变位后圆柱体的总的储油量的时候也应该进行分段讨论，且根据题意可知此时分段应建立在求解变位后圆柱体部分的储油量分段的基础上，即：

（1）当0?h0?d2tan?时，由图12可以看出液面还未到达右边的球冠体部分，所以变位后圆柱体的总的储油量为

V4?V3?V左 （5.2.16）

h0?r?ltan??h0其中，V3??tan?dl??r2r2?x2dx。

?d1（2）当d2t由图13可以看出液面均到达了左右两边的球an?﹤h0?2r-d1tan?时，

冠体，所以可得变位后圆柱体的总的储油量为

V4?V3?V左?V右 （5.2.17）

6dl?r?ltan??x2r2?x2dx。 其中，V3???2??r（3）当2r-d1tan??h0?2r时，由图14可以看出液面已经充满了左边的球冠体，可以计算出左边的球冠体的总容积为

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V左???R?(R2?x2)dx

R?d3其中，R为实际圆柱型油罐两端球冠的半径。

右边的球冠体仍可按以上情况解决，所以可得变位后圆柱体的总的储油量为：

V4?V3?V左??V右 （5.2.18）

h?2r?r?ltan??h0d其中，V3??r2(0)??2dl??r2r2?x2dxa；

h0?2rtan?tan?V左?为左边的球冠体的总容积，V左半部球冠体的储油量；V右为右半部球冠体的

左储油量。

4、考虑到圆柱油罐的横向倾斜时，其示意图如下所示：

?r 液面 h0 h

图19 圆柱油罐的横向倾斜示意图

由于实际的圆筒油罐的在游位探针出的横截面为圆形，则当油桶发生横向倾斜时，油桶内液面到油桶最底部的高度不会发生变化，但是游位探针及其上面的油浮子会随着油桶发生横向倾斜，则油位探针显示的油位高度h将会随着?角的变化发生变化。

则由几何关系式可得油面实际高度h0和油位探针显示高度h间的关系为 h0?r?(r?h)cos? （5.2.18） 其中，r为油罐圆柱体横截面半径。 5.2.2模型的求解

综合上面的模型得出油罐的总体积为：V4?V3?V左?V右 ， 其中，V右?VGHE?VEFH； VGHE=2?rr?h222dx?0r?x[R2?x2?y2?(R?d3)]dy；

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r?h2r2?x22?x2?y2?(R?d)?(x?h?r)cot?]dyVEFH=2?[R?320x2

6dl?r?ltan??x2r2?x2dx； V3???2??rV左?VBCD+VABC；

h?rr2?x2[R2?x2?y2?(R?d)]dy； VBCD?2??1dx?03r2?x2x1rVABC?2?dx?0[R2?x2?y2?(R?d3)??x?h1?r?cot?]dy h1?r将其代入在Matlab7.0的平台下进行编程（程序见附表5），可以得到含参变量纵向倾斜角度?和横向偏转角度?体积V关与油面高度的函数，这里用V?F(h,?,?)表示，具体表达式见附表6。

要求得参数?,? 这里采用最小二乘法[3]来求，问题即转化为解最优化问题，算法的基本思路如下:

由于在油罐变位后附件2中的显示储油量不是实际的储油量，而每次的出油量为实际值。故利用出油后显示的油位高度和出油量来拟合求出参量?,?使得理论出油量值与实际出油量值之查的平方和最小如下所示:

min?,?1?v(h,?,?)?vdata222?12(?v(hdatai,?,?)?vdatai)， ?2i其中， ?v?F(hdatai?1,?,?)?F(hdatai,?,?)

hdatai为实际数据中采集到的各流水号对应的油位高；

vdatai为实际数据中采集到的各流水号对应的出油量；其中i为对应的流水号。

将给定的数据带入程序(见附录7)运用Matlab7.0中的最小二乘函数工具[4] lsqcurvefit可得到参数的值：??2.12°；??4.38°。

将得到??2.12°、??4.38°带入V?F(h,?,?)求解出?v相对于vdata的残差平方和[5]为2.926?10?4。由于残差平方和比较小，则认为函数V?F(h,?,?)在参数??2.12°、??4.38°情况下可以较好的反映油位高度与油罐储油量的相对关系。此时V?F(h)函数见附录8

基于在Matlab7.0平台上运行程序可以求得罐体变位后油位高度间隔为10cm的罐容表标定值如下表：

表20 油位高度间隔为10cm的罐容表标定值

油位高度/m 储油量/m3 油位高度/m 储油量/m3 油位高度/m 储油量/m3

0.0 0.0510 1.1 19.2557 2.2 49.3141 0.1 0.3550 1.2 21.9311 2.3 51.7684 0.2 1.0643 1.3 24.6647 2.4 54.1023

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0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.2193 3.6974 5.4261 7.3639 9.4797 11.7478 14.1462 16.655 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 27.4406 30.2431 33.0570 35.8670 38.6577 41.4138 44.1194 46.7585 2.5 2.6 2.7 2.8 2.9 3 56.2949 58.3225 60.1570 61.7629 63.0882 64.0216

对模型进一步讨论可得，随着参数?、?的变化，油罐实际储油量和显示的油位高关系式也讲发生变化。下面列出若干参数?、?的值，作出油罐储油量和油位高的关系式V?F(h,?,?)的函数图，用于识别油罐的变位程度。

图21 油罐的变位程度识辨图

由上图可以看出当参数?、?取不同值时，即油罐变位程度不同时油位高度与油罐储油量的相对关系明显不同。当参数?、?趋近于0时函数V?F(h)与油罐变位后显示数据的残差平方和为7.7332?10?5，残差平方和比较小，又因油罐变位后显示数据关系与油罐无变位时相等，所以说明所建立的模型函数在参数

?、?趋近于0时与油罐无变位时的关系比较接近，即进一步说明所建立的模型函数可以较好的反映油罐变位后油位高度与储油量的响度关系。

六、模型的评价及推广

6.1模型的评价

文中建立的模型较为理想地解决了此类问题，根据几何知识基础利用微积分中曲面积分和二重积分的方法在理论上求得椭圆油罐和实际圆柱型油罐在变位前后的储油量与实际油位指针显示的油位高度之间的关系，并且经过检验理论值和实验值的误差也在允许的范围之内，最后也给出了变位之后的罐容表标值。

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同时，模型也存在一些缺点和不足：

（1）问题一中，在对理论值进行修正时，不能考虑液体对油罐内壁压力对油罐形状的影响，也没有考虑到实验时会有油吸附在油罐内壁上，导致理论值和实验值仍存在一定的误差。

（2）问题二中，在分别对圆柱部分和两端的球冠体进行二重积分时，由于各种变量之间的代换之间十分复杂，导致用计算机软件求积分时结果很麻烦，而且求解出的圆柱型油罐的储油量和油位高之间的关系式十分冗长，不利于之后求解过程的进行。 6.2模型的推广

（1）利用微积分知识求解油罐储油量的方法可以推广到实际问题中形状更为复杂的油罐及其变位后储油量的计算，可以为工业生产中求容器中液体的溶液体积的值提供了理论依据。

（2）在求解参数a和?的过程中使用的最小二乘法可以推广应用到参数估计，也可应用最小二乘法辨别闭环系统，以及推广到人口预测模型中求未知参数的值

参考文献：

[1] 姜启源，谢金星，叶俊，数学模型（第3版），北京:高等教育出版社，2003.8 [2] 周义仓,贺效良,数学建模实验,西安：西安交通大学出版社,1999.3

[3] 代冬岩，李智勇，张宏礼，最小二乘曲线拟合及其MATLAB实现，科学信息技术报，第29卷第2期：140-143，2000 [4] 最小二乘曲线拟合Matlab的最小二乘法

http://www.24u88.cn/viewnews-2869.html，2010.9.13 [5] 最小二乘最优问题—残差的检验

http://moonlab.net/log/nonlinear-least-squares.html，2010.9.13

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附录

附录1

syms x l t; %第一问几何求解函数 y=0.89*2*sqrt(1-t^2/0.36) a=int(int(y,t,-0.6,-l*0.07168+x-0.6),l,-0.4,2.05) b=int(int(y,t,-0.6,-l*0.07168+x-0.6),l,-0.4,x/0.07168) c=int(int(y,t,-0.6,-l*0.07168+x-0.6),l,(x-1.2)/0.07168,2.05)+1.67761*((x-1.2)/0.07168+0.4) 附录2 a =

89/49218750000000*(-48321616-244140625*x^2+364718750*x)^(3/2)+13083/20000*pi-267/560000*(-48321616-244140625*x^2+364718750*x)^(1/2)+3116157/560000*asin(-11671/9375+5/3*x)-6675/896*asin(-11671/9375+5/3*x)*x-89/49218750000000*(8199296-244140625*x^2+278968750*x)^(3/2)+267/560000*(8199296-244140625*x^2+278968750*x)^(1/2)-2383509/560000*asin(-8927/9375+5/3*x)+6675/896*asin(-8927/9375+5/3*x)*x b =

6675/1792*pi*x-2383509/1120000*pi-89/49218750000000*(8199296-244140625*x^2+278968750*x)^(3/2)+267/560000*(8199296-244140625*x^2+278968750*x)^(1/2)-2383509/560000*asin(-8927/9375+5/3*x)+6675/896*asin(-8927/9375+5/3*x)*x c =

89/49218750000000*(-48321616-244140625*x^2+364718750*x)^(3/2)+8122407/1120000*pi-267/560000*(-48321616-244140625*x^2+364718750*x)^(1/2)+3116157/560000*asin(-11671/9375+5/3*x)-6675/896*asin(-11671/9375+5/3*x)*x-6675/1792*pi*x+11805130891989090625/504403158265495552*x-34569200893629493859/1261007895663738880

附录3 wc.m x=0:0.01:0.1469; %函数a段 y=6675/1792*pi*x-2383509/1120000*pi-89/49218750000000*(278968750*x-244140625*x.^2+8199296).^(3/2)+267/560000*(278968750*x-244140625*x.^2+8199296).^(1/2)-2383509/560000*asin(-8927/9375+5/3*x)+6675/896*asin(-8927/9375+5/3*x).*x x1=0.1469:0.01:1.1713; %函数b段 y1=89/49218750000000*(-48321616-244140625*x1.^2+364718750*x1).^(3/2)+13083/20000*pi-267/560000*(-48321616-244140625*x1.^2+364718750*x1).^(1/2)+3116157/560000*asin(-11671/9375+5/3*x1)-6675/896*asin(-11671/9375+5/3*x1).*x1-89/49218750000000*(278968750*x1-244140625*x1.^2+8199296).^(3/2)+267/560000*(278968750*x1-244140625*x1.^2+8199296).^(1/2)-238

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3509/560000*asin(-8927/9375+5/3*x1)+6675/896*asin(-8927/9375+5/3*x1).*x1 x2=1.1713:0.01:1.2; %函数c段 y2=89/49218750000000*(-48321616-244140625*x2.^2+364718750*x2).^(3/2)+8122407/1120000*pi-267/560000*(-48321616-244140625*x2.^2+364718750*x2).^(1/2)+3116157/560000*asin(-11671/9375+5/3*x2)-6675/896*asin(-11671/9375+5/3*x2).*x2-6675/1792*pi*x2+11805130891989090625/504403158265495552*x2-34569200893629493859/1261007895663738880 %附件1中罐体变位进油的油位及油体积的数据 x3=[0.41129 0.42345 0.43833 0.45054 0.4639 0.47774 0.48937 0.50256 0.51469 0.52684 0.53888 0.55196 0.5644 0.57656 0.58874 0.59956 0.61162 0.62344 0.63558 0.64628 0.65859 0.67022 0.68063 0.69303 0.70467 0.71645 0.72766 0.73939 0.7509 0.76155 0.77343 0.78539 0.79604 0.80827 0.8208 0.8328 0.84447 0.85629 0.8676 0.88006 0.89292 0.90434 0.91734 0.9299 0.94142 0.9546 0.96809 0.98014 0.99241 1.00634 1.01907 1.03424 1.03536] y3=[0.96286 1.01286 1.06286 1.11286 1.16286 1.21286 1.26286 1.31279 1.36279 1.41273 1.46273 1.51273 1.56273 1.61273 1.66273 1.71273 1.76273 1.81273 1.86273 1.91273 1.96273 2.01273 2.06273 2.11273 2.16273 2.21273 2.26273 2.31273 2.36273 2.41273 2.46273 2.51273 2.56273 2.61273 2.66273 2.71273 2.76273 2.81273 2.86273 2.91273 2.96273 3.01273 3.06273 3.11273 3.16273 3.21273 3.26273 3.31273 3.36273 3.41273 3.46273 3.51273 3.51474] %附件1中罐体变位出油的油位及油体积的数据 x4=[1.02065 1.00773 0.99432 0.98096 0.9671 0.95601 0.94154 0.92969 0.91644 0.90414 0.8919 0.87923 0.86899 0.85513 0.84402 0.83164 0.82047 0.80816 0.796 0.78504 0.77307 0.76209 0.75081 0.73942 0.72709 0.71532 0.70543 0.69352 0.6825 0.67102 0.65868 0.64774 0.63576 0.62461 0.61253 0.60069 0.5894 0.577 0.56458 0.55433 0.54076 0.52865 0.51719 0.50487 0.49078 0.47806 0.46597 0.4524 0.43998 0.42583 0.41173] y4=[3.46474 3.41474 3.36474 3.31474 3.26474 3.21474 3.16474 3.11474 3.06474 3.01474 2.96474 2.91474 2.86474 2.81474 2.76474 2.71474 2.66474 2.61474 2.56474 2.51474 2.46474 2.41474 2.36474 2.31474 2.26474 2.21474 2.16474 2.11474 2.06474 2.01474 1.96474 1.91474 1.86474 1.81474 1.76474 1.71474 1.66474 1.61474 1.56474 1.51474 1.46474 1.41474 1.36474 1.31474 1.26474 1.21474 1.16474 1.11474 1.06474 1.01474 0.96474] %模型函数与罐体实际进油数据的最大误差 y5=89/49218750000000*(-48321616-244140625*x3.^2+364718750*x3).^(3/2)+13083/20000*pi-267/560000*(-48321616-244140625*x3.^2+364718750*x3).^(1/2)+3116157/560000*asin(-11671/9375+5/3*x3)-6675/896*asin(-11671/9375+5/3*x3).*x3-89/49218750000000*(278968750*x3-244140625*x3.^2+8199296).^(3/2)+267/560000*(278968750*x3-244140625*x3.^2+8199296).^(1/2)-2383509/560000*asin(-8927/9375+5/3*x3)+6675/896*asin(-8927/9375+5/3*x3).*x3 t=(y5-y3) l=max(abs(t))

22

%模型函数与罐体实际出油数据的最大误差 y6=89/49218750000000*(-48321616-244140625*x4.^2+364718750*x4).^(3/2)+13083/20000*pi-267/560000*(-48321616-244140625*x4.^2+364718750*x4).^(1/2)+3116157/560000*asin(-11671/9375+5/3*x4)-6675/896*asin(-11671/9375+5/3*x4).*x4-89/49218750000000*(278968750*x4-244140625*x4.^2+8199296).^(3/2)+267/560000*(278968750*x4-244140625*x4.^2+8199296).^(1/2)-2383509/560000*asin(-8927/9375+5/3*x4)+6675/896*asin(-8927/9375+5/3*x4).*x4 p=(y6-y4)./y6 k=max(abs(p)) 附录 4 grb.m x=0:0.01:0.14; %模型函数1段 y=6675/1792*pi*x-2383509/1120000*pi-89/49218750000000*(278968750*x-244140625*x.^2+8199296).^(3/2)+267/560000*(278968750*x-244140625*x.^2+8199296).^(1/2)-2383509/560000*asin(-8927/9375+5/3*x)+6675/896*asin(-8927/9375+5/3*x).*x x1=0.15:0.01:0.41; %模型函数2段 y1=3.0811*x1-0.3044 x2=0.42:0.01:1.03; %模型函数3段 y2=89/49218750000000*(-48321616-244140625*x2.^2+364718750*x2).^(3/2)+13083/20000*pi-267/560000*(-48321616-244140625*x2.^2+364718750*x2).^(1/2)+3116157/560000*asin(-11671/9375+5/3*x2)-6675/896*asin(-11671/9375+5/3*x2).*x2-89/49218750000000*(278968750*x2-244140625*x2.^2+8199296).^(3/2)+267/560000*(278968750*x2-244140625*x2.^2+8199296).^(1/2)-2383509/560000*asin(-8927/9375+5/3*x2)+6675/896*asin(-8927/9375+5/3*x2).*x2-(-0.3974*x2.^2+0.5834*x2-0.1242) x3=1.04:0.01:1.17; %模型函数4段 y3=3.2781*x3+0.1208 x4=1.18:0.01:1.2; %模型函数5段 y4=89/49218750000000*(-48321616-244140625*x4.^2+364718750*x4).^(3/2)+8122407/1120000*pi-267/560000*(-48321616-244140625*x4.^2+364718750*x4).^(1/2)+3116157/560000*asin(-11671/9375+5/3*x4)-6675/896*asin(-11671/9375+5/3*x4).*x4-6675/1792*pi*x4+11805130891989090625/504403158265495552*x4-34569200893629493859/1261007895663738880 %几何求出函数求解罐容表 y' size(y') y1' size(y1') y2' size(y2') y3'

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size(y3') y4' size(y4') 附录5 zzz.m syms l h0 x h1 y alpha hh x1 xx; y1=int(int(2*sqrt(2.25-x^2),x,-1.5,-1.5-l*tan(alpha)+h0),l,-2,6) y2=int(int(2*(sqrt(1.625^2-x^2-y^2)-0.625),y,0,sqrt(2.25-x^2)),x,-1.5,h1-1.5) y3=int(int(2*(sqrt(1.625^2-x^2-y^2)-0.625-(x-h1+1.5)*cot(alpha)),y,0,sqrt(2.25-x^2)),x,h1-1.5,x1) y4=int(int(2*(sqrt(1.625^2-x^2-y^2)-0.625),y,0,sqrt(2.25-x^2)),x,1.5-hh,1.5) y5=int(int(-2*(sqrt(1.625^2-x^2-y^2)-0.625-(x+hh-1.5)*cot(alpha)),y,0,sqrt(2.25-x^2)),x,1.5-hh,xx) y=y1+y2+y3+y4+y5 附录6

y=31/24*pi-9/4*cot(alpha)*(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*asin(2/3*(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-1)+3*cot(alpha)*(-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).^2+3*(2*tan(alpha)+1.5-(1.5-h)*cos(beta))).^(1/2).*(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+9/4*cot(alpha)*(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*asin(2/3*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha))))-3/4*cot(alpha)*(9-4*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha))).^2).^(1/2).*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha)))+9/4*cot(alpha)*(-6*tan(alpha)+1.5-(1.5-h)*cos(beta)).*asin(2/3*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(

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1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))))-3/4*cot(alpha)*(9-4*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))).^2).^(1/2).*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha)))-9/4*cot(alpha)*(-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).^2+3*(2*tan(alpha)+1.5-(1.5-h)*cos(beta))).^(1/2)+2/3*cot(alpha)*(3*(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))-(-6*tan(alpha)+1.5-(1.5-h)*cos(beta)).^2).^(3/2)+169/64*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha))).*atan(4/5*(9-4*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha))).^2).^(1/2))-1/3*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha))).^3.*atan(4/5*(9-4*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha))).^2).^(1/2))+1/3*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alph

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a)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))).^3.*atan(4/5*(9-4*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))).^2).^(1/2))-1/12*cot(alpha)*(9-4*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))).^2).^(3/2)-27/8*cot(alpha)*asin(2/3*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))))-169/64*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))).*atan(4/5*(9-4*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))).^2).^(1/2))-2197/1536*atan(2/5*(18+13*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(a

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lpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))))./(9-4*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))).^2).^(1/2))-2/3*cot(alpha)*(-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).^2+3*(2*tan(alpha)+1.5-(1.5-h)*cos(beta))).^(3/2)+1205/768*asin(2/3*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))))+9/4*cot(alpha)*(3*(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))-(-6*tan(alpha)+1.5-(1.5-h)*cos(beta)).^2).^(1/2)+1/2*cot(alpha)*(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha))).*(9-4*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha))).^2).^(1/2)-27/8*cot(alpha)*asin(-1+2/3*(-6*tan(alpha)+1.5-(1.5-h)*cos(beta)))-1205/768*asin(2/3*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha))))-5/24*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)

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)-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha))).*(9-4*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha))).^2).^(1/2)+5/24*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))).*(9-4*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))).^2).^(1/2)-cot(alpha)*(-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).^2+3*(2*tan(alpha)+1.5-(1.5-h)*cos(beta))).^(1/2).*(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).^2+27/8*cot(alpha)*asin(2/3*(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-1)-2197/1536*atan(2/5*(-18+13*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))))./(9-4*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))).^2).^(1/2))+1/12*cot(alpha)*(9-4*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)

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*cos(beta))))/2/(1/sin(alpha)/sin(alpha))).^2).^(3/2)-27/8*cot(alpha)*asin(2/3*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha))))+cot(alpha)*(3*(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))-(-6*tan(alpha)+1.5-(1.5-h)*cos(beta)).^2).^(1/2).*(-6*tan(alpha)+1.5-(1.5-h)*cos(beta)).^2-3*cot(alpha)*(3*(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))-(-6*tan(alpha)+1.5-(1.5-h)*cos(beta)).^2).^(1/2).*(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))+1/2*cot(alpha)*(-6*tan(alpha)+1.5-(1.5-h)*cos(beta)).*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))).*(9-4*((-(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))+sqrt((2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))).*(2*cot(alpha)*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)))-4*(1/sin(alpha)/sin(alpha))*((0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha)).*(0.625-1.5*cot(alpha)+(-6*tan(alpha)+1.5-(1.5-h)*cos(beta))*cot(alpha))-1.625*1.625)))/2/(1/sin(alpha)/sin(alpha))).^2).^(1/2)-1/24*(54*(12*tan(alpha)*(1.5-(1.5-h)*cos(beta))-36*tan(alpha)^2-(1.5-(1.5-h)*cos(beta)).^2+3*(1.5-(1.5-h)*cos(beta))-18*tan(alpha)).^(1/2)+54*asin(-1-4*tan(alpha)+2/3*(1.5-(1.5-h)*cos(beta))).*(1.5-(1.5-h)*cos(beta))-8*(12*tan(alpha)*(1.5-(1.5-h)*cos(beta))-36*tan(alpha)^2-(1.5-(1.5-h)*cos(beta)).^2+3*(1.5-(1.5-h)*cos(beta))-18*tan(alpha)).^(3/2)-324*asin(-1-4*tan(alpha)+2/3*(1.5-(1.5-h)*cos(beta)))*tan(alpha)-81*asin(-1-4*tan(alpha)+2/3*(1.5-(1.5-h)*cos(beta)))-216*pi*tan(alpha)-54*(-4*tan(alpha)*(1.5-(1.5-h)*cos(beta))-4*tan(alpha)^2-(1.5-(1.5-h)*cos(beta)).^2+3*(1.5-(1.5-h)*cos(beta))+6*tan(alpha)).^(1/2)-54*asin(-1+4/3*tan(alpha)+2/3*(1.5-(1.5-h)*cos(beta))).*(1.5-(1.5-h)*cos(beta))+8*(-4*tan(alpha)*(1.5-(1.5-h)*cos(beta))-4*tan(alpha)^2-(1.5-(1.5-h)*cos(beta)).^2+3*(1.5-(1.5-h)*cos(beta))+6*tan(alpha)).^(3/2)-108*asin(-1+4/3*tan(alpha)+2/3*(1.5-(1.5-h)*cos(beta)))*tan(alpha)+81*asin(-1+4/3*tan(alpha)+2/3*(1.5-(1.5-h)*cos(beta))))/tan(alpha)+2197/1536*atan(2/5*(-18+13*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alph

29

a)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha))))./(9-4*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha))).^2).^(1/2))+2197/1536*atan(2/5*(18+13*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha))))./(9-4*((-(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))+sqrt((2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta)).*(2*cot(alpha)*(0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))-4*(1/sin(alpha)/sin(alpha))*((0.625+1.5*cot(alpha)-(2*tan(alpha)+1.5-(1.5-h)*cos(beta))))/2/(1/sin(alpha)/sin(alpha))).^2).^(1/2))+9/4*cot(alpha)*(-6*tan(alpha)+1.5-(1.5-h)*cos(beta)).*asin(-1+2/3*(-6*tan(alpha)+1.5-(1.5-h)*cos(beta)))

附录7 ccc.m h=[2.63223 2.6243 2.62067 2.61029 2.60661 2.59959 2.5876 2.58205 2.57957 2.57544 2.56946 2.56412 2.55983 2.54847 2.53963 2.52801 2.52163 2.51023 2.50817 2.50007 2.49006 2.48573 2.4744 2.46477 2.45451 2.44677 2.43685 2.43155 2.42732 2.4222 2.41435 2.40405 2.39915 2.39312 2.3825 2.37435 2.36244 2.3584 2.34813 2.33937 2.33488 2.32813 2.32214 2.31414 2.30414 2.30109 2.29087 2.28046 2.27492 2.26861 2.26089 2.25188 2.24246 2.23288 2.22699 2.2207 2.20913 2.2014 2.19091 2.18614 2.17792 2.17004 2.1584 2.15554 2.14854 2.14132 2.13842 2.12737 2.11653 2.11014 2.10032 2.09684 2.08864 2.08403 2.07758 2.06714 2.06317 2.05814 2.0513 2.04592 2.03594 2.02406 2.02047 2.01611 2.00708 2.00133 1.98959 1.97787 1.96943 1.95883 1.95281 1.94449 1.93264 1.92505 1.91371 1.90451 1.89767 1.88927 1.8784 1.87441 1.86846 1.85654 1.85051 1.84192 1.83091 1.81896 1.81042 1.80734 1.80498 1.7968 1.78913 1.77751 1.76805 1.75942 1.75219 1.74759 1.73597 1.72857 1.72626 1.7173 1.7101 1.70751 1.69661 1.69131 1.68701 1.68387 1.67877 1.67448 1.66596 1.6633 1.65855 1.65373 1.64293 1.63648 1.62692 1.61889 1.60906 1.60592 1.59413 1.58365 1.58114 1.57448 1.56922 1.56092 1.55662 1.54882 1.54079 1.53279 1.5263 1.52395 1.51681 1.51073 1.50765 1.50106 1.49455 1.48703 1.47698 1.46797 1.45725 1.45473 1.45053 1.44393 1.43235 1.42245 1.41593 1.4106 1.40801 1.3986 1.39153 1.38753 1.38126 1.37757 1.36805 1.36237 1.35095 1.34878 1.33849 1.33022 1.32284 1.31433 1.30506 1.30212 1.29134 1.2892 1.28426 1.28046 1.26919 1.26651 1.2587 1.25033 1.24182 1.23117 1.22861 1.21844 1.21115 1.20221 1.19809

30

1.19065 1.18163 1.17006 1.16362 1.16077 1.15819 1.1499 1.13994 1.13102 1.12557 1.1141 1.1069 1.09536 1.09263 1.08856 1.07881 1.06766 1.05784 1.05288 1.04936 1.03889 1.02904 1.02433 1.02005 1.01484 1.00454] y=[60.44888 60.31143 60.24803 60.06511 59.99969 59.87406 59.65702 59.55551 59.50994 59.43377 59.32285 59.22317 59.14266 58.92769 58.75861 58.53401 58.40958 58.18531 58.14452 57.98336 57.78253 57.69508 57.46467 57.26702 57.05465 56.89324 56.68486 56.57286 56.48312 56.37411 56.20614 55.98422 55.87805 55.74687 55.51445 55.3349 55.07068 54.98057 54.7504 54.55285 54.45117 54.29775 54.16106 53.97771 53.74727 53.67672 53.43937 53.19616 53.06614 52.91756 52.73508 52.52114 52.29637 52.06665 51.92485 51.77296 51.49232 51.30395 51.0472 50.93004 50.72753 50.53269 50.24362 50.17237 49.99762 49.81682 49.74405 49.46596 49.19194 49.02986 48.77999 48.69121 48.48156 48.36342 48.19779 47.92887 47.82634 47.69624 47.51896 47.37923 47.11935 46.80888 46.71483 46.60046 46.36309 46.21159 45.90147 45.5908 45.36643 45.08388 44.92305 44.70035 44.38233 44.17812 43.87231 43.6236 43.43833 43.21041 42.91484 42.80616 42.64393 42.31832 42.1533 41.91787 41.61555 41.2867 41.05126 40.96626 40.90109 40.67504 40.46279 40.14074 39.87812 39.63821 39.437 39.30887 38.98485 38.77825 38.71371 38.46322 38.26174 38.18922 37.88379 37.73515 37.61449 37.52635 37.38313 37.2626 37.02308 36.94826 36.81461 36.67894 36.37474 36.19294 35.92333 35.69673 35.41917 35.33047 34.9973 34.70098 34.62999 34.44158 34.29275 34.05783 33.9361 33.71525 33.48784 33.26123 33.07738 33.0108 32.80851 32.63623 32.54896 32.36223 32.17783 31.96475 31.68 31.42474 31.1211 31.04973 30.93079 30.74392 30.41613 30.13601 29.95159 29.80087 29.72764 29.46168 29.26194 29.14898 28.97196 28.86781 28.59924 28.43909 28.1173 28.05619 27.76655 27.53396 27.32655 27.08757 26.82749 26.74505 26.44303 26.38311 26.24487 26.13858 25.82364 25.74882 25.53091 25.29763 25.06073 24.76466 24.69357 24.4114 24.20941 23.96204 23.84816 23.64272 23.394 23.07557 22.89862 22.82038 22.7496 22.52238 22.24989 22.00633 21.85774 21.5456 21.35006 21.03734 20.96348 20.85345 20.59032 20.29019 20.02655 19.89365 19.79943 19.51972 19.25731 19.13209 19.01844 18.88029 18.60778] a0=[0.045 0.045] a=lsqcurvefit(@myfun,a0,h,y) 调用函数myfun.m如下：

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myfun.m function F=myfun(a,h) h=[h 0.41398]; F1=31/24*pi-9/4*cot(a(1))*(2*tan(a(1))+1.5-(1.5-h)*cos(a(2))).*asin(2/3*(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))-1)+3*cot(a(1))*(-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2))).^2+3*(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))).^(1/2).*(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))+9/4*cot(a(1))*(2*tan(a(1))+1.5-(1.5-h)*cos(a(2))).*asin(2/3*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))))-3/4*cot(a(1))*(9-4*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^2).^(1/2).*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1))))+9/4*cot(a(1))*(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2))).*asin(2/3*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))))-3/4*cot(a(1))*(9-4*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^2).^(1/2).*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)

32

)))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1))))-9/4*cot(a(1))*(-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2))).^2+3*(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))).^(1/2)+2/3*cot(a(1))*(3*(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))-(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2))).^2).^(3/2)+169/64*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).*atan(4/5*(9-4*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^2).^(1/2))-1/3*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^3.*atan(4/5*(9-4*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^2).^(1/2))+1/3*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^3.*atan(4/5*(9-4*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1)

33

)+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^2).^(1/2))-1/12*cot(a(1))*(9-4*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^2).^(3/2)-27/8*cot(a(1))*asin(2/3*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))))-169/64*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).*atan(4/5*(9-4*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^2).^(1/2))-2197/1536*atan(2/5*(18+13*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))))./(9-4*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos

34

(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^2).^(1/2))-2/3*cot(a(1))*(-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2))).^2+3*(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))).^(3/2)+1205/768*asin(2/3*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))))+9/4*cot(a(1))*(3*(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))-(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2))).^2).^(1/2)+1/2*cot(a(1))*(2*tan(a(1))+1.5-(1.5-h)*cos(a(2))).*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).*(9-4*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^2).^(1/2)-27/8*cot(a(1))*asin(-1+2/3*(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2))))-1205/768*asin(2/3*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))))-5/24*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).*(9-4*((-(2*cot(a(1))*(

35

0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^2).^(1/2)+5/24*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).*(9-4*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^2).^(1/2)-cot(a(1))*(-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2))).^2+3*(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))).^(1/2).*(2*tan(a(1))+1.5-(1.5-h)*cos(a(2))).^2+27/8*cot(a(1))*asin(2/3*(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))-1)-2197/1536*atan(2/5*(-18+13*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))))./(9-4*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^2).^(1/2))+1/12*cot(a(1))*(9-4*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.62

36

5)))/2/(1/sin(a(1))/sin(a(1)))).^2).^(3/2)-27/8*cot(a(1))*asin(2/3*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))))+cot(a(1))*(3*(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))-(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2))).^2).^(1/2).*(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2))).^2-3*cot(a(1))*(3*(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))-(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2))).^2).^(1/2).*(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))+1/2*cot(a(1))*(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2))).*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).*(9-4*((-(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625-1.5*cot(a(1))+(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^2).^(1/2)-1/24*(54*(12*tan(a(1))*(1.5-(1.5-h)*cos(a(2)))-36*tan(a(1))^2-(1.5-(1.5-h)*cos(a(2))).^2+3*(1.5-(1.5-h)*cos(a(2)))-18*tan(a(1))).^(1/2)+54*asin(-1-4*tan(a(1))+2/3*(1.5-(1.5-h)*cos(a(2)))).*(1.5-(1.5-h)*cos(a(2)))-8*(12*tan(a(1))*(1.5-(1.5-h)*cos(a(2)))-36*tan(a(1))^2-(1.5-(1.5-h)*cos(a(2))).^2+3*(1.5-(1.5-h)*cos(a(2)))-18*tan(a(1))).^(3/2)-324*asin(-1-4*tan(a(1))+2/3*(1.5-(1.5-h)*cos(a(2))))*tan(a(1))-81*asin(-1-4*tan(a(1))+2/3*(1.5-(1.5-h)*cos(a(2))))-216*pi*tan(a(1))-54*(-4*tan(a(1))*(1.5-(1.5-h)*cos(a(2)))-4*tan(a(1))^2-(1.5-(1.5-h)*cos(a(2))).^2+3*(1.5-(1.5-h)*cos(a(2)))+6*tan(a(1))).^(1/2)-54*asin(-1+4/3*tan(a(1))+2/3*(1.5-(1.5-h)*cos(a(2)))).*(1.5-(1.5-h)*cos(a(2)))+8*(-4*tan(a(1))*(1.5-(1.5-h)*cos(a(2)))-4*tan(a(1))^2-(1.5-(1.5-h)*cos(a(2))).^2+3*(1.5-(1.5-h)*cos(a(2)))+6*tan(a(1))).^(3/2)-108*asin(-1+4/3*tan(a(1))+2/3*(1.5-(1.5-h)*cos(a(2))))*tan(a(1))+81*asin(-1+4/3*tan(a(1))+2/3*(1.5-(1.5-h)*cos(a(2)))))/tan(a(1))+2197/1536*atan(2/5*(-18+13*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(

37

a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))))./(9-4*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^2).^(1/2))+2197/1536*atan(2/5*(18+13*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))))./(9-4*((-(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))+sqrt((2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))).*(2*cot(a(1))*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))))-4*(1/sin(a(1))/sin(a(1)))*((0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1))).*(0.625+1.5*cot(a(1))-(2*tan(a(1))+1.5-(1.5-h)*cos(a(2)))*cot(a(1)))-1.625*1.625)))/2/(1/sin(a(1))/sin(a(1)))).^2).^(1/2))+9/4*cot(a(1))*(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2))).*asin(-1+2/3*(-6*tan(a(1))+1.5-(1.5-h)*cos(a(2)))) F2=F1(2:end); F1(end)=[]; F=F1-F2 附录8

y=10.291666666666666666666666666667*pi-20.261019425943518759951793881972*(9-4*(-1.4427492525329528170562069124258+.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^2)^(1/2)*(-1.4427492525329528170562069124258+.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))+81.044077703774075039807175527889*(-(.784093782015238530424896799543e-1+.99708293931141132209973125814771*h)^2+.235228134604571559127469039862

38

9+2.9912488179342339662991937744431*h)^(1/2)*(.784093782015238530424896799543e-1+.99708293931141132209973125814771*h)+(-2.9408973767218486197439854932607+13.467944535110244919797186250155*h)*(1.6922713764343777178508559146839-.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))*(9-4*(1.6922713764343777178508559146839-.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^2)^(1/2)-92.743597833412501086449739135542*asin(-.96183283502196854470413794161720+.66381237036786111958663631519205*h+.45612513122926715370589876663770e-3*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))-20.261019425943518759951793881972*(9-4*(1.6922713764343777178508559146839-.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^2)^(1/2)*(1.6922713764343777178508559146839-.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))-(-13.234038195248318788847934719673+60.605750407996102139087338125696*h)*asin(-1.1281809176229184785672372764559+.66381237036786111958663631519205*h-.45612513122926715370589876663770e-3*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))-(4.7659618047516812111520652803274+60.605750407996102139087338125696*h)*asin(-.94772708119898409797167354669713+.66472195954094088139982083876514*h)+(4.7659618047516812111520652803274+60.605750407996102139087338125696*h)*asin(-.96183283502196854470413794161720+.66381237036786111958663631519205*h+.45612513122926715370589876663770e-3*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))+60.783058277830556279855381645917*asin(-.94772708119898409797167354669711+.664721959540940881399820

39

83876514*h)*(.43755910328830168504031127784e-2+.99708293931141132209973125814771*h)+(1.0591026232781513802560145067394+13.467944535110244919797186250155*h)*(-1.4427492525329528170562069124258+.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))*(9-4*(-1.4427492525329528170562069124258+.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^2)^(1/2)+89.605577000079167753116405802208*asin(-1.1281809176229184785672372764559+.66381237036786111958663631519205*h-.45612513122926715370589876663770e-3*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))+(-13.234038195248318788847934719673+60.605750407996102139087338125696*h)*asin(-1.1451505136486929944839043924996+.66472195954094088139982083876514*h)-60.783058277830556279855381645917*(-.70056267675031010428792939915996+3.4341557746450170850675526069854*h-(.43755910328830168504031127784e-2+.99708293931141132209973125814771*h)^2)^(1/2)-9.0048975226415638933119083919877*(.22909924981150900141465536697508+2.8436131656973062600430741635956*h-(.43755910328830168504031127784e-2+.99708293931141132209973125814771*h)^2)^(3/2)+27.014692567924691679935725175963*(-.6531773114191184751775697662482+2.9912488179342339662991937744431*h-(-.2177257704730394917258565887494+.99708293931141132209973125814771*h)^2)^(1/2)*(-.2177257704730394917258565887494+.99708293931141132209973125814771*h)^2-(4.4686541033970286611999163997122-2.6293193107539499033626922797060*h+.18066831369784253666319583959790e-2*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))*atan(4/5*(9-4*(1.6922713764343777178508559146839-.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^2)^(1/2))-(-.30057276094436517022004310675537+.20744136573995659987082384849752*h+.14253910350914598553309336457428e-3*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))*(9-4*(-1.4427

40

492525329528170562069124258+.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^2)^(1/2)-81.044077703774075039807175527889*(-.6531773114191184751775697662482+2.9912488179342339662991937744431*h-(-.2177257704730394917258565887494+.99708293931141132209973125814771*h)^2)^(1/2)*(-.2177257704730394917258565887494+.99708293931141132209973125814771*h)-86.674587416745834419783072468876*asin(-.94772708119898409797167354669711+.66472195954094088139982083876514*h)+104.67458741674583441978307246888*asin(-1.1451505136486929944839043924995+.66472195954094088139982083876514*h)-60.783058277830556279855381645917*(-(.784093782015238530424896799543e-1+.99708293931141132209973125814771*h)^2+.2352281346045715591274690398629+2.9912488179342339662991937744431*h)^(1/2)+2197/1536*atan((-.30229611317135464869227594461400+5.1777364888693167327757632584980*h+.35577760235882837989060103797741e-2*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))/(9-4*(-1.4427492525329528170562069124258+.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^2)^(1/2))-2197/1536*atan((1.5998111574587641328244507563564-5.1777364888693167327757632584980*h+.35577760235882837989060103797741e-2*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))/(9-4*(1.6922713764343777178508559146839-.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^2)^(1/2))+2.2512243806603909733279770979969*(9-4*(-1.4427492525329528170562069124258+.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^2)^(3/2)-2.2512243806603909733279770979969*(9-4*(1.6922713764343777178508559146839-.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.17445815794228958123124240

41

26*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^2)^(3/2)+(.35255653675716202455226164889248-.20744136573995659987082384849752*h+.14253910350914598553309336457428e-3*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))*(9-4*(1.6922713764343777178508559146839-.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^2)^(1/2)+18.009795045283127786623816783975*(-.6531773114191184751775697662482+2.9912488179342339662991937744431*h-(-.2177257704730394917258565887494+.99708293931141132209973125814771*h)^2)^(3/2)+1/3*(1.6922713764343777178508559146839-.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^3*atan(4/5*(9-4*(1.6922713764343777178508559146839-.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^2)^(1/2))+2197/1536*atan((-14.702296113171354648692275944614+5.1777364888693167327757632584980*h+.35577760235882837989060103797741e-2*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))/(9-4*(-1.4427492525329528170562069124258+.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^2)^(1/2))-1/3*(-1.4427492525329528170562069124258+.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^3*atan(4/5*(9-4*(-1.4427492525329528170562069124258+.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.93588907022048983959437250030

42

9*h)^2+7719.0075535733163584254387494694)^(1/2))^2)^(1/2))-27.014692567924691679935725175963*(-(.784093782015238530424896799543e-1+.99708293931141132209973125814771*h)^2+.2352281346045715591274690398629+2.9912488179342339662991937744431*h)^(1/2)*(.784093782015238530424896799543e-1+.99708293931141132209973125814771*h)^2+(-3.8097597449698285325390463781244+2.6293193107539499033626922797060*h+.18066831369784253666319583959790e-2*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))*atan(4/5*(9-4*(-1.4427492525329528170562069124258+.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((2108.7038822653952991872576765733-1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(39.028833605330734759391558750465-26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^2)^(1/2))-91.174587416745834419783072468875*asin(-1.1451505136486929944839043924996+.66472195954094088139982083876514*h)+60.783058277830556279855381645917*(-.6531773114191184751775697662482+2.9912488179342339662991937744431*h-(-.2177257704730394917258565887494+.99708293931141132209973125814771*h)^2)^(1/2)-2197/1536*atan((15.999811157458764132824450756356-5.1777364888693167327757632584980*h+.35577760235882837989060103797741e-2*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))/(9-4*(1.6922713764343777178508559146839-.99571855555179167937995447278808*h+.68418769684390073055884814995655e-3*((-2473.4022319323786368663899664489+1455.3295245514588008713872805381*h)^2-2923.1744581579422895812312424026*(-45.778833605330734759391558750465+26.935889070220489839594372500309*h)^2+7719.0075535733163584254387494694)^(1/2))^2)^(1/2))+9.0048975226415638933119083919877*(-.70056267675031010428792939915996+3.4341557746450170850675526069854*h-(.43755910328830168504031127784e-2+.99708293931141132209973125814771*h)^2)^(3/2)-60.783058277830556279855381645917*asin(-1.1451505136486929944839043924995+.66472195954094088139982083876514*h)*(.43755910328830168504031127784e-2+.99708293931141132209973125814771*h)+91.174587416745834419783072468875*asin(-.94772708119898409797167354669713+.66472195954094088139982083876514*h)+60.783058277830556279855381645917*(.22909924981150900141465536697508+2.8436131656973062600430741635956*h-(.43755910328830168504031127784e-2+.99708293931141132209973125814771*h)^2)^(1/2)-18.009795045283127786623816783975*(-(.784093782015238530424896799543e-1+.99708293931141132209973125814771*h)^2+.2352281346045715591274690398629+2.9912488179342339662991937744431*h)^(3/2)

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