Integration wt parametric f*2

註: 省略a倍, 最後體積再乘以 a^3即可
Q1. Rotate about the x-axis(disk method)
dx=d(θ+sinθ) = (1+cosθ) dθ
V= 2∫[0~π] π(1+cosθ)^2 *(1+cosθ)dθ
= 2∫[0~π] π[1+3cosθ+3(1+cos2θ)/2 + (1-sin^2 θ)cosθ] dθ
= 2π{(5/2)θ+3sinθ+(3/4)sin(2θ)+[sinθ- (1/3)(sinθ)^3]}, θ=0~π
= 5π^2
Ans: V=5π^2*a^3
Q2: Rotate about the y-axis(shell method)
V=∫[0~π] 2π(θ+sinθ)*(1+cosθ)^2 dθ
= 2π∫[0~π] [θ(1+cosθ)^2+ sinθ(1+cosθ)^2] dθ
= 2π[(A)+(B)]
(A)=∫[0~π] θ[1+2cosθ+0.5(1+cos2θ)]dθ (integration by parts)
={θ[(3θ)/2+2sinθ+(1/4)sin(2θ)]-(3/4)θ^2+2cosθ+(1/8)cos(2θ)}|[0~π]
= (3π^2)/2 - (3π^2)/4- 4=(3π^2)/4- 4
(B)=∫[0~π] sinθ(1+cosθ)^2 dθ
= (-1/3)(1+cosθ)^3 , θ=0~π
= 8/3
V=2π[(A)+(B)]=2π[(3/4)π^2 - 4/3] = π(9π^2 -16)/6
Ans: πa^3(9π^2- 16)/6