Chapter 17.4, Problem 42E

To determine

(a) The proof that circle generated by rotating a point $(0,a,b)$ about the $z-axis$ is parameterized by $(a\cos \theta ,a\sin \theta ,b),(0\le \theta \le 2\pi )$.

(b) The proof that $S$ is parameterized by $G(y,\theta )=(y\cos \theta ,y\sin \theta ,g(y)),for(c\le y\le d,0\le \theta \le 2\pi )$.

(c) The proof of $Area(S)=2\pi {\displaystyle \underset{c}{\overset{d}{\int }}y}\sqrt{1+{\left[g\text{'}(y)\right]}^2}dy$.