计算二重积分:xsin(x+y)dσ , D:0≤x≤π,0≤y≤π/2

题目

计算二重积分:xsin(x+y)dσ , D:0≤x≤π,0≤y≤π/2
急,希望详细一点,谢谢!

答案

两重积分先积sin(x+y)dy
II xsin(x+y)dσ=Ixdx Isin(x+y)dy
=I xdx I sin(x+y)d(x+y)
=I xdx(-cos(x+y)):pi/2-0
=I xdx(-cos(x+pi/2)+cos(x))
=I xcos(x)dx- I xcos(x+pi./2)dx
=(xsinx- I sinxdx) - I (x+pi/2)cos(x+pi/2)dx+pi/2 * I cos(x+pi/2)dx
=cosx -(x+pi/2)sin(x+pi/2)-cos(x+pi/2)+pi/2*sin(x+pi/2):pi-0
=-2-(-3pi/2-pi/2)-0+pi/2*-2
=-2+2pi-pi
=pi-2