Line integral with polar coordinates

Question

A curve C in polar form $r=f(\theta)$ is parameterized by $C(\theta) = \langle f(\theta)\cos(\theta),f(\theta) \sin(\theta)\rangle$ because the x- and y-coordinates are given by $x = r \cos(\theta)$ and $y=r \sin(\theta)$. Evaluate $\int_C(x-y)^2ds$ where C is given by $r= 2 \cos(\theta)$ , $0\le \theta\le\frac{\pi}{2}$.

I'm not sure I'm understanding this question. If $r= 2 cos(\theta)$, then $C(\theta) = \langle2\cos^2(\theta),2\cos(\theta) \sin(\theta)\rangle$

Since $C'(\theta) = \langle -4\cos(\theta)\sin(\theta), 2\cos(2\theta)\rangle$, $||C'(\theta)|| = 2$. The integral is set up like this: $$2\int_0^{\frac{\pi}{2}}(2\cos^2(\theta)-2\cos(\theta)\sin(\theta))^2d\theta$$

This is not an easy integral to do, unless I am missing something. Any tips?

Answer 1

$$I=\int_0^{\frac{\pi}{2}}(2\cos^2\theta-\sin2\theta)^2d\theta$$ $$I=4\int_0^{\frac{\pi}{2}}\cos^4\theta d\theta+\int_0^{\frac{\pi}{2}}\sin^22\theta\,d\theta-8\int_0^{\frac{\pi}{2}}\cos^3\theta\sin\theta \,d\theta$$ use $$\sin^22\theta=\frac{1-\cos4\theta}{2}$$ $$\int_{0}^{\frac{\pi}{2}}\sin^{2x-1}\theta\cos^{2y-1}\theta\,d\theta=\frac12\beta(x,y)$$ Indeed $$\int_0^{\frac{\pi}{2}}\cos^4\theta d\theta=\frac12\beta(\frac 12,\frac 52)$$ and $$\int_0^{\frac{\pi}{2}}\cos^3\theta\sin\theta \,d\theta=\frac12\beta( 1,2)$$

Other way $$\cos^3\theta \sin\theta=(1-\sin^2\theta) \cos\theta \sin \theta$$ set $u=\sin\theta$ and

$$\cos^4\theta=\frac{1}{4}(1+2\cos2\theta)+\frac{1}{8}(1+\cos4\theta)$$