Question

Find the surface area of the solid formed by the rotation of an arc of the cycloid
x= a(θ+sinθ), y = a(1+cosθ) about the axis of x.

Answer 1

$x(\theta)=a(\theta+sin(\theta))\\ y(\theta)=a(1+sin(\theta))\\$

area calculation formula:

$P=2\pi \displaystyle\intop _{\theta_1} ^{\theta_2} y(\theta)*\sqrt{(x'(\theta))^2+(y'(\theta))^2}d\theta\\ x'(\theta)=a+a*cos(\theta)\\ y'(\theta)=-asin(\theta)\\$

after substitution and simplification

$P=2\pi\displaystyle\intop _{0} ^{\pi} a*(1+cos(\theta))*a*2*cos(\theta/2)d\theta=\\ =4\pi a^2\displaystyle\intop_{0}^{\pi}(cos(\theta/2)+cos(\theta)*cos(\theta/2))d\theta=\\ =32\pi a^2/3$

since we calculated only half of the figure, the result should be multiplied by 2

answer: $64 \pi *a^2/3$