计算二重积分∫∫D（siny/y）dxdy,其中D是由直线y=x和抛物线x=y^2所围城的区域.

题目

计算二重积分∫∫D（siny/y）dxdy,其中D是由直线y=x和抛物线x=y^2所围城的区域.

答案

交点为(0,0),(1,1)
V = ∫(0->1)∫(y²->y) siny/y dxdy
= ∫(0->1) xsiny/y |(y²->y) dy
= ∫(0->1) (y-y²)siny/y dy
= ∫(0->1) (1-y)siny dy
= ∫(0->1) (y-1) d(cosy)
= (y-1)cosy |(0->1) - ∫(0->1) cosy d(y-1)
= -[(-1)(1)] - siny |(0->1)
= 1 - (sin(1) - 0)
= 1 - sin(1)