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

∫ f ( a ) f ( b ) 2 π f ( y ) 1 + f ′ ( y ) 2 d y .

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.