A region between two parabolas — AP® Calculus AB

Name: ______________________ Date: ____________ A region between two parabolas (9 points)

Graph

Let

f(x) = 6x - x^2

and

g(x) = x^2

. Let

R

be the region enclosed by the graphs of

f

and

g

, as shown. No calculator is allowed on this question.

(a) Find the area of $R$ . [2 points]
(b) Find the volume of the solid generated when $R$ is revolved about the horizontal line $y = -1$ . [3 points]
(c) Region $R$ is the base of a solid. For this solid, each cross section perpendicular to the $x$ -axis is a semicircle whose diameter lies in $R$ . Write, but do not evaluate, an integral expression for the volume of the solid. [2 points]
(d) Write, but do not evaluate, an integral expression for the volume of the solid generated when $R$ is revolved about the vertical line $x = -1$ . (For $0 \le x \le 3$ , the graph of $f$ is also the graph of $x = 3 - \sqrt{9 - y}$ .) [2 points]

The question