定积分之旋转曲面的面积

极坐标

设AB⌢ \stackrel \frown {AB} 的方程为ρ=ρ(θ)\rho=\rho(\theta)，α≤θ≤β\alpha≤\theta≤\beta,则

S=2π∫βαρ(θ)sinθ[ρ(θ)]2+[ρ′(θ)]2−−−−−−−−−−−−−√dθS=2\pi \int_\alpha ^\beta \rho(\theta)\sin \theta \sqrt {[\rho(\theta)]^2+[\rho'(\theta)]^2} d\theta

例：求心形线(心脏线)ρ=a(1+cosθ)\rho=a(1+\cos \theta),a>0a>0绕极轴旋转一周的旋转曲面面积。

代公式

弧元ds=[ρ(θ)]2+[ρ′(θ)]2−−−−−−−−−−−−−√dθ=[a(1+cosθ)]2+[a(−sinθ)]2−−−−−−−−−−−−−−−−−−−−−−√dθ=2√a1+cosθ−−−−−−−√dθds=\sqrt {[\rho(\theta)]^2+[\rho'(\theta)]^2} d\theta=\sqrt {[a(1+\cos \theta)]^2+[a(-\sin \theta)]^2} d\theta=\sqrt 2a \sqrt {1+\cos \theta} d\theta

S=2π∫βαρ(θ)sinθ[ρ(θ)]2+[ρ′(θ)]2−−−−−−−−−−−−−√dθS=2\pi \int_\alpha ^\beta \rho(\theta)\sin \theta \sqrt {[\rho(\theta)]^2+[\rho'(\theta)]^2} d\theta

=2π∫π0a(1+cosθ)sinθ2√a1+cosθ−−−−−−−√dθ=2\pi \int_0 ^\pi a(1+\cos \theta)\sin \theta \sqrt 2a \sqrt {1+\cos \theta} d\theta

=22√πa2∫π0(1+cosθ)32sinθdθ=2\sqrt 2\pi a^2 \int_0 ^\pi (1+\cos \theta)^{\frac{3}{2}}\sin \theta d\theta

=−22√πa2∫π0(1+cosθ)32d(1+cosθ)=-2\sqrt 2\pi a^2 \int_0 ^\pi (1+\cos \theta)^{\frac{3}{2}}d(1+\cos \theta)

=−22√πa225(1+cosθ)52|π0=-2\sqrt 2\pi a^2 \frac{2}{5}(1+\cos \theta)^{\frac{5}{2}}|_0^{\pi}

=−22√πa225(−42√)=-2\sqrt 2\pi a^2 \frac{2}{5}(-4\sqrt 2)

=325πa2= \frac{32}{5}\pi a^2

心形线matlab画法,matlab中可以使用polar极坐标画图

θ为0→π\theta为0\to\pi的心形线

clear
clc
theta=0:pi/100:pi;
r=4*(1-cos(theta));
polar(theta,r)

θ为0→2π\theta为0\to2\pi的心形线

clear
clc
theta=0:pi/100:2*pi;
r=4*(1-cos(theta));
polar(theta,r)