高数题求详解，旋转体的体积怎么求的
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(1) 绕 y=-1,V1 = ∫ π[(x+1)^2-(x^2+1)^2]dx
= ∫ π(2x-x^2-x^4)dx = π[x^2-x^3/3-x^5/5] = 7π/15.
(2) 绕 x=-1,V2 = ∫ π[(√y+1)^2-(y+1)^2]dy
= ∫ π(2√y-y-y^2)dx = π[(4/3)y^(3/2)-y^2/2-y^3/3] = π/2.
或用柱壳法,V2 = ∫ 2π(x+1)(x-x^2)dx
= ∫ 2π(x-x^3)dx = π[x^2-x^4/2] = π/2
= ∫ π(2x-x^2-x^4)dx = π[x^2-x^3/3-x^5/5] = 7π/15.
(2) 绕 x=-1,V2 = ∫ π[(√y+1)^2-(y+1)^2]dy
= ∫ π(2√y-y-y^2)dx = π[(4/3)y^(3/2)-y^2/2-y^3/3] = π/2.
或用柱壳法,V2 = ∫ 2π(x+1)(x-x^2)dx
= ∫ 2π(x-x^3)dx = π[x^2-x^4/2] = π/2
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