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二重积分的黎曼和

2009-08-11 22:55:53| 分类：高等数学图形动画 | 标签： |举报 |字号大中小订阅

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高等数学图形与动画

二重积分的黎曼和

在引入二重积分的定义之前，教材给出了一个引例：用n个很细的平顶柱体的体积之和去逼近曲顶柱体的体积。
平顶柱体的体积之和就是二重积分的黎曼和（Riemann sum）。
以下动画形象地演示了这个逼近过程：

with(plots):with(plottools):
f:=(x,y)->10-x^2-y^2:
a:=-2:b:=2:c:=-2:d:=2:
N:=20:for n from 1 to N do
for i from 0 to n do xi:=a+i*(b-a)/n:
for j from 0 to n do yj:=c+j*(d-c)/n:
B[n,i,j]:=polygonplot3d({
[[xi,yj,f(xi,yj)],[xi+(b-a)/n,yj,f(xi,yj)],[xi+(b-a)/n,yj+(d-c)/n,f(xi,yj)],[xi,yj+(d-c)/n,f(xi,yj)]],
[[xi,yj,0],[xi+(b-a)/n,yj,0],[xi+(b-a)/n,yj,f(xi,yj)],[xi,yj,f(xi,yj)]],
[[xi+(b-a)/n,yj,0],[xi+(b-a)/n,yj+(d-c)/n,0],[xi+(b-a)/n,yj+(d-c)/n,f(xi,yj)],[xi+(b-a)/n,yj,f(xi,yj)]],
[[xi+(b-a)/n,yj+(d-c)/n,0],[xi,yj+(d-c)/n,0],[xi,yj+(d-c)/n,f(xi,yj)],[xi+(b-a)/n,yj+(d-c)/n,f(xi,yj)]],
[[xi,yj,0],[xi,yj+(d-c)/n,0],[xi,yj+(d-c)/n,f(xi,yj)],[xi,yj,f(xi,yj)]],
[[xi,yj,0],[xi+(b-a)/n,yj,0],[xi+(b-a)/n,yj+(d-c)/n,0],[xi,yj+(d-c)/n,0]]}) od:
A[n,i]:=display(seq(B[n,i,j],j=0..n-1)) od:
AA[n]:=display(seq(A[n,i],i=0..n-1)) od:
AAA:=display(seq(AA[n],n=1..N),insequence=true):
display(AAA,AAA,axes=boxed,tickmarks=[0,0,0],orientation=[54,54]);

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