Chapter 10.1, Problem 80ES

If $f^{\prime }\left(t\right)\text{ and }{g}^{\prime }\left(t\right)$ are continuous functions, and if no segment of the curve $x=f\left(t\right),y=g\left(t\right)\left(a\le t\le b\right)$ is traced more than once, then it can be shown that the area of the surface generated by revolving this curve about the x-axis is $S={\displaystyle {\int }_a^{b}2\pi y\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt}$ and the area of the surface generated by revolving the curve about the y-axis is $S={\displaystyle {\int }_a^{b}2\pi x\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt}$ [The derivations are similar to those used to obtain Formulas (4) and (5) in Section 6.5.] Use the formulas above in these exercises.

The equations $x=a\varphi -a\sin \varphi ,y=a-a\cos \varphi \left(0\le \varphi \le 2\pi \right)$ represent one arch of a cycloid. Show that the surface area generated by revolving this curve about the x-axis is given by $S=64\pi a^2/3.$