Chapter 6.6, Problem 21E

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

1. a. If the curve y = f(x) on the interval [a, b] is revolved about the y-axis, the area of the surface generated is

${\displaystyle {\int }_{f(a)}^{f(b)}2\pi f(y)\sqrt{1+f^{\prime }{(y)}^2}dy.}$

1. b. If f is not one-to-one on the interval [a, b] then the area of the surface generated when the graph of f on [a, b] is revolved about the x-axis is not defined.
2. c. Let f(x) = 12x². The area of the surface generated when the graph of f on [−4, 4] is revolved about the x-axis is twice the area of the surface generated when the graph of f on [0, 4] is revolved about the x-axis.
3. d. Let f(x) = 12x². The area of the surface generated when the graph of f on [−4, 4] is revolved about the y-axis is twice the area of the surface generated when the graph of f on [0, 4] is revolved about the y-axis.