Problem

A curve has parametric equations

$$x=a(\theta-\sin\theta) \qquad y=a(1-\cos\theta)$$

where $a$ is a positive constant.

(a) Show that

$$\left(\frac{dx}{d\theta}\right)^2+\left(\frac{dy}{d\theta}\right)^2 =ka^2\sin^2\frac{\theta}{2}$$

where $k$ is a constant to be determined.

The part of the curve from $\theta=0$ to $\theta=2\pi$ is rotated through $2\pi$ radians about the $x$-axis.

(b) Determine the area of the surface generated, giving your answer in terms of $\pi$ and $a$.

[Solutions relying on calculator technology are not acceptable.]