求心形线P=a(1+cost)绕极轴旋转所得旋转体的体积

Question

求心形线P=a(1+cost)绕极轴旋转所得旋转体的体积

Answer 1

由极坐标下曲线ρ=ρ(θ)绕极轴旋转所得的体积可以用以极点O为顶点,极径ρ为母线的圆锥体积增量来积分.以ρ=ρ(θ)为母线的圆锥的体积为V(ρ,θ)=(π/3)(ρsinθ)^2(ρcosθ)=(π/3)ρ^3(sinθ)^2cosθ将ρ=a(1+cosθ)代入上式,可得：V(ρ,θ)=V(θ)=(π/3)a^3(1+cosθ)^3(sinθ)^2cosθ令F(θ)=(1+cosθ)^3(sinθ)^2cosθ,则V(θ)=(1/3)πa^3F(θ)从而V(θ+dθ)=(1/3)πa^3F(θ+dθ),可得：dV=V(θ+dθ)-V(θ)=[dV(θ)/dθ]dθ当圆锥的顶角大于π/2时,V(θ+dθ)