2010年我校参加全国大学生数学建模竞赛获全国二等奖论文A题

d5=int(f6,-1,t);

v312=[v312 eval(d5)];

%求圆柱体积

f3=((1.5^2)/2*asin((-y*tan(a*pi/180)+h1(i)-1.5)/1.5)+((-y*tan(a*pi/180)+h1(i)-1.5)/2)*sqrt(1.5^2-(-y*tan(a*pi/180)+h1(i)-1.5)^2)+pi*1.5^2/4)*2;

d3=int(f3,0,8); v32=[v32 eval(d3)];

%求水平面与右球冠的交点

f4=(y-7.375)^2+(-y*tan(a*pi/180)+h1(i)-1.5)^2-1.625^2; x00=solve(f4);

y00=max(eval(x00));

%求右球冠体积

f5=((R^2-(y-7.375)^2)/2*asin((h1(i)-y*tan(a*pi/180)-1.5)/(sqrt(R^2-(y-7.375)^2)))+((h1(i)-y*tan(a*pi/180)-1.5)/2)*sqrt(R^2-(y-7.375)^2/2-(h1(i)-y*tan(a*pi/180)-1.5)^2)+pi*(R^2-(y-7.375)^2)/4)*2; y00

d4=int(f5,8,y00); v33=[v33 eval(d4)]; end

v31=v311+v312; v3=v31+v32+v33;

disp('第三部分的体积'); v3'

%第四部分体积

h=[1.9:0.1:2.8 2.63]; h1=h+2*tan(a*pi/180); y0=[];

%求水平面与左球冠的交点 for i=1:length(h1)

f1=(y-0.625)^2+(-y*tan(a*pi/180)+h1(i)-1.5)^2-1.625^2; x0=solve(f1);

y0=[y0 min(eval(x0))]; end %y0

v411=[]; v412=[];

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v41=[]; v42=[]; v431=[]; v432=[]; v43=[]; v4=[];

for i=1:length(h1) %求左球冠体积

f2=((R^2-(0.625-y)^2)/2*asin((h1(i)-y*tan(a*pi/180)-1.5)/(sqrt(R^2-(0.625-y)^2)))+((h1(i)-y*tan(a*pi/180)-1.5)/2)*sqrt(R^2-(0.625-y)^2-(h1(i)-y*tan(a*pi/180)-1.5)^2)+pi*(R^2-(0.625-y)^2)/4)*2; t=y0(i);

d2=int(f2,t,0);

v411=[v311 eval(d2)];

f6=pi*(R^2-(0.625-y)^2); t=y0(i);

d5=int(f6,-1,t);

v412=[v312 eval(d5)]; v41=v411+v412;

%求圆柱体积

f3=((1.5^2)/2*asin((-y*tan(a*pi/180)+h1(i)-1.5)/1.5)+((-y*tan(a*pi/180)+h1(i)-1.5)/2)*sqrt(1.5^2-(-y*tan(a*pi/180)+h1(i)-1.5)^2)+pi*1.5^2/4)*2;

d3=int(f3,0,8); v42=[v42 eval(d3)];

%求水平面与右球冠的交点

f4=(y-7.375)^2+(-y*tan(a*pi/180)+h1(i)-1.5)^2-1.625^2; x00=solve(f4);

y00=max(eval(x00));

%求右球冠体积

f5=((R^2-(y-7.375)^2)/2*asin((h1(i)-y*tan(a*pi/180)-1.5)/(sqrt(R^2-(y-7.375)^2)))+((h1(i)-y*tan(a*pi/180)-1.5)/2)*sqrt(R^2-(y-7.375)^2/2-(h1(i)-y*tan(a*pi/180)-1.5)^2)+pi*(R^2-(y-7.375)^2)/4)*2; d4=int(f5,8,y00); v431=[v431 eval(d4)];

f9=pi*(R^2-(y-7.375)^2);

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d9=int(f9,y00,9); v432=[v432 eval(d9)]; end

v43=v431+v432; v4=v41+v42+v43;

disp('第四部分的体积'); v4'

%第五部分体积 h=2.9:0.1:3;

h1=h+2*tan(a*pi/180); y0=[];

%求水平面与左球冠的交点 for i=1:length(h1)

f1=(y-0.625)^2+(-y*tan(a*pi/180)+h1(i)-1.5)^2-1.625^2; x0=solve(f1);

y0=[y0 min(eval(x0))]; end %y0

v51=[]; v52=[]; v53=[]; v541=[]; v542=[]; v54=[]; v5=[];

for i=1:length(h1) %求左球冠体积

f6=pi*(R^2-(0.625-y)^2); d5=int(f6,-1,0); v51=[v51 eval(d5)];

%求圆柱体积

f4=(y-7.375)^2+(-y*tan(a*pi/180)+h1(i)-1.5)^2-1.625^2;%求水平面与右球冠的交点

x00=solve(f4);

y00=max(eval(x00));

v52=[v52 pi*1.5^2*y00];

f3=((1.5^2)/2*asin((-y*tan(a*pi/180)+h1(i)-1.5)/1.5)+((-y*tan(a*pi/180)+h1(i)-1.5)/2)*sqrt(1.5^2-(-y*tan(a*pi/180)+h1(i)-1.5)^2)+pi*1.5^2/

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4)*2;

d3=int(f3,y00,8); v53=[v53 eval(d3)];

%求右球冠体积

f5=((R^2-(y-7.375)^2)/2*asin((h1(i)-y*tan(a*pi/180)-1.5)/(sqrt(R^2-(y-7.375)^2)))+((h1(i)-y*tan(a*pi/180)-1.5)/2)*sqrt(R^2-(y-7.375)^2/2-(h1(i)-y*tan(a*pi/180)-1.5)^2)+pi*(R^2-(y-7.375)^2)/4)*2; d4=int(f5,8,y00); v541=[v541 eval(d4)];

f9=pi*(R^2-(y-7.375)^2); d9=int(f9,y00,9); v542=[v542 eval(d9)]; end

v54=v541+v542;

v5=v51+v52+v53+v54; disp('第五部分的体积'); v5'

附录四：

用来求纵向和横向同时变位的第一部分的体积：

syms y;

a=3;R=1.625;b=4.3;

%第一部分体积 h=0:0.1:0.3;

h1=h*cos(b*pi/180)+1.5*(1-cos(b*pi/180))+2*tan(a*pi/180); y0=[];

for i=1:length(h1)

f1=(y-0.625)^2+(-y*tan(a*pi/180)+h1(i)-1.5)^2-1.625^2; x0=solve(f1);

y0=[y0 min(eval(x0))]; end y0'

v11=[]; v12=[];

for i=1:length(h1) %求左球冠体积

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