Question

计算二重积分∫∫xydσ 其中D是由曲线y=x 2及直线x=1,y=0轴围成的闭区域

Answer 1

{ y = x²、y = 0
{ x = 1
∫∫ xy dxdy
= ∫[0→1] dx ∫[0→x²] xy dy
= ∫[0→1] x * [y²/2]：[0→x²] dx
= ∫[0→1] x/2 * x⁴ dx
= ∫[0→1] (1/2)x⁵ dx
= (1/2)(1/6)x⁶：[0→1]
= 1/12

可以说简单画个图就解决了。