1.∫∫D（x²-y²)dxdy
=∫(0,π)dx∫(0,sinx)（x²-y²)dy
=∫(0,π)dx[x²sinx - (sin³x)/3]
=∫(0,π)[x²(-dcosx) + (sin²x)(dcosx)/3]
=∫(0,π)[(1-cos²x)(dcosx)/3] - x²cosx|(0,π) + ∫(0,π)cosx dx²
=(3cosx-cos³x)/9|(0,π) - x²cosx|(0,π) + 2∫(0,π)x dsinx
=π² - 4/9 + 2xsinx|(0,π) - 2∫(0,π)sinx dx
=π² - 4/9 + 2cosx|(0,π)
=π² - 40/9
2.所围的区域是1/2 < y < 2,1/y < x < 2
∫∫Dye^xydxdy
=∫(1/2,2)ydy∫(1/y,2)e^xydx
=∫(1/2,2)ydy * [e^(2y) - e]/y
=∫(1/2,2)dy * [e^(2y) - e]
=[e^(2y)/2 - ey]|(1/2,2)
=(e^4)/2 - 2e
3.交线是x²+2y²=6-2x²-y²,即x²+y²=2
体积=∫∫D(z1-z2)dxdy
=∫∫D[6-2x²-y²-(x²+2y²)]dxdy
用极坐标代换,令x=rcost,y=rsint
则0