(1) ∫[0,1]dy∫[y,1]e^xdx
=∫[0,1](e-e^y)dy
=e-e^y[0,1]
=1
(2) 原式=∫[0,π/2]dθ∫[0,1]ρ^4sin^2(θ)cosθdρ
=∫[0,π/2]sin^2(θ)cosθdθ∫[0,1]ρ^4dρ
=1/3sin^3(θ)[0,π/2]*1/5ρ^5[0,1]
=1/15
(3) 此题有问题区域不封闭
(4) 原式=∫[0,2π]dθ∫[0,2]ρ(4-ρ^2)dρ
=2π*(2ρ^2-1/4ρ^4)[0,2]
=2π*(8-4)
=8π
(5) 原式=∫[0,π]dx∫[x^2,πx]sinx/xdy
=∫[0,π]sinx/x(πx-x^2)dx
=π∫[0,π]sinxdx- ∫[0,π]xsinxdx
=-πcosx[0,π]+ xcosx[0,π]-π∫[0,π]cosxdx
=2π-π
=π