Chapter 11.1, Problem 53ES

(a) The accompanying figure shows a surface of revolution that is generated by revolving the curve $y=f\left(x\right)$ in the xy-plane about the x-axis. Show that the equation of this surface is $y^2+z^2\text{=}{\text{[}f\left(x\right)]}^2.$

(b) Find an equation of the surface of revolution that is generated by revolving the curve $y=e^x$ in the xy-plane about the x-axis.

(c) Show that the ellipsoid $3x^2+4y^2+4z^2=16$ is a surface of revolution about the x-axis by finding a curve $y=f\left(x\right)$ in the xy-plane that generates it.