求∫∫D(x*2－y*2)?dxdy,D是以O(0,0),A(1,-1),B(1,1)为顶点的三角形,

题目

求∫∫D(x*2－y*2)?dxdy,D是以O(0,0),A(1,-1),B(1,1)为顶点的三角形,

答案

在三角形区域 -x<=y<=x
∫∫D(x*2－y*2)?dxdy
=∫[0,1]dx∫[-x,x] (x^2-y^2)dy
=∫[0,1]dx (x^2y-1/3y^3)[-x,x]
=∫[0,1]4/3x^3dx
=1/3x^4 [0.1]
=1/3