Question

Hello everyone!
How do I find the volume by rotation of 1 arc of the cycloid: x=a(t-sint), y=a(1-cost)?

Answer 1

What's the axis of revolution?

Answer 2

Actually, it is not stated, but i suppose it is the x axis.

Answer 3

Looking at the plot of these parametric functions, you can see that one arc of the cycloid is completed when \(y(t)=0\). The simplest starting point would be \(t=0\), since \(y(0)=a(1-\cos0)=0\), followed by the next best choice of \(t=2\pi\), since \(y(2\pi)=a(1-\cos2\pi)=0\).

The volume of the revolved region is given by
\[\pi\int_0^{2\pi}y(t)^2 \frac{\mathrm{d}x}{\mathrm{d}t}\,\mathrm{d}t=\pi a^3\int_0^{2\pi}(1-\cos t)^2\sin t\,\mathrm{d}t\]