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计算二重积分:xsin(x+y)Dσ , D:0≤x≤π,0≤y≤π/2

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/...

x、y互相独立,可以分离;

xsin(x+y)dxdy 在积分区域D上的积分 = {[xsin(x+y)dy从0到(派/2)的积分]dx从0到(派)的积分}, [xsin(x+y)dy从0到(派/2)的积分] =[-xcos(x+y)]在y=派/2处的值-[-xcos(x+y)]在y=0处的值 =-xcos(x+派/2)+xcosx=xsinx+xcosx, 所以xsin(x+y)dxdy在D上...

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