分からない問題はここに書いてね464

>>988
8(c) V = {(x, y, z) ; x^2+y^2 ≦ ax, x^2+y^2+z^2 ≦ a^2 }
x = r cosθ, y = r sinθ → dxdy = r drdθ
V = {(r, θ, z) ; 0 ≦ r ≦ a cosθ, z^2 ≦ a^2 - r^2 }
a ≧ 0 なら cosθ ≧ 0 だから -π/2 ≦ θ ≦ π/2
V = ∫_V dxdydz = ∫_V r drdθdz
= ∫_(-π/2 ≦ θ ≦ π/2) ∫_(0 ≦ r ≦ a cosθ) ∫_(-√(a^2 - r^2) ≦ z ≦ √(a^2 - r^2)) r dzdrdθ
= ∫_(-π/2 ≦ θ ≦ π/2) ∫_(0 ≦ r ≦ a cosθ) 2r√(a^2 - r^2) drdθ
= ∫_(-π/2 ≦ θ ≦ π/2) [ -(2/3)(a^2 - r^2)^(3/2) ]_(0 ≦ r ≦ a cosθ) dθ
= ∫_(-π/2 ≦ θ ≦ π/2) ( (2/3) a^3 - (2/3) a^3 (sinθ)^3 ) dθ
= (2/3) a^3 (π - 2)