A-A+
估计下列各二重积分的值：xsin(x+y)dσ 其中区域D：0≤x≤π 0≤y≤π．
问题详情
估计下列各二重积分的值：xsin(x+y)dσ，其中区域D：0≤x≤π，0≤y≤π．
参考答案
两重积分先积sin(x+y)dyIIxsin(x+y)dσ=IxdxIsin(x+y)dy=IxdxIsin(x+y)d(x+y)=Ixdx(-cos(x+y)):pi/2-0=Ixdx(-cos(x+pi/2)+cos(x))=Ixcos(x)dx-Ixcos(x+pi./2)dx=(xsinx-Isinxdx)-I(x+pi/2)cos(x+pi/2)dx+pi/2*Icos(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
