Answer to Question #115710 in Calculus for Navjot

Question

Find the volume of solid obtained by revolving one arch of cycloid x=a(theta+sin theta) and y=a(1-cos theta)

Answer 1

The line of cycloids is set parametrically. When it comes to the first arch, then there is a limitation:

$0<0<2\pi$ and then the first arch is drawn when the parameter value changes within: $0\le t \le 2\pi$. We use the formula for finding the body volume formed by a line given parametrically: $V=\pi \int_a^bf^2(t)x\prime(t)dt$. Substitute ve into the formula and get: $V=\pi\int^{2\pi}_0(a(1-\cos{\theta})(a(\theta+\sin{\theta})^\prime d\theta=\pi\int^{2\pi}_0(a(1-\cos{\theta})(\theta+\cos{\theta})d\theta=\pi\int^{2\pi}_0(a\theta-a\cos^2{\theta}+a\cos{\theta}- ax\cos{\theta)}d\theta=-a\pi\int^{2\pi}_0(\frac{1}{2}\cos{2\theta}+\frac{1}{2})d\theta-a\pi\int^{2\pi}_0\theta\cos{\theta}d\theta+a\pi\int^{2\pi}_0\cos{\theta}d\theta+a\pi\int^{2\pi}_0\theta d\theta=\pi^2(2\pi-1)a$