计算二重积分:∫∫D cos(x+y)dxdy,其中D由y=x,y=π,x=0所围成的区域

题目

计算二重积分:∫∫D cos(x+y)dxdy,其中D由y=x,y=π,x=0所围成的区域

答案

∫∫_D cos(x + y) dσ
= ∫(0→π) dy ∫(0→y) cos(x + y) dx
= ∫(0→π) dy ∫(0→y) cos(x + y) d(x + y)
= ∫(0→π) sin(x + y) |(0→y) dy
= ∫(0→π) [sin(2y) - sin(y)] dy
= cos(y) - (1/2)cos(2y) |(0→π)
= [cos(π) - (1/2)cos(2π)] - [cos(0) - (1/2)cos(0)]
= - 2