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!

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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!
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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!
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!

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!
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

先计算y的积分,把x当已知,再计算x的积分,用分部积分法就可以了

∫∫xsin(x+y)dxdy D:0≤x≤π,0≤y≤π/2
=∫(0,π)dx∫(0,π/2)xsin(x+y)dy
=-∫(0,π)xcos(x+y)|(0,π/2)dx
=-[∫(0,π)xcos(x+π/2)dx-∫(0,π)xcosxdx]
=-[xsin(x+π/2)|(0,π)+cos(x+π/2)|(0,π)-xsinx|(0,π)-cosx|(0,π)]
=-(-π+2)
=π-2