### Home > APCALC > Chapter 8 > Lesson 8.2.1 > Problem8-71

8-71.

Multiple Choice: The volume generated by revolving the region enclosed by the graphs of $y = 2x$ and $y = 2x^2$ for $0 ≤ x ≤ 1$ about the $x$-axis is:

1. $\pi \int _ { 0 } ^ { 1 } ( 2 x - 2 x ^ { 2 } ) ^ { 2 } d x$

1. $\pi \int _ { 0 } ^ { 1 } ( 4 x ^ { 2 } - 4 x ^ { 4 } ) d x$

1. $\pi \int _ { 0 } ^ { 2 } ( \sqrt { \frac { y } { 2 } } - \frac { y } { 2 } ) ^ { 2 } d x$

1. $2 \pi \int _ { 0 } ^ { 1 } x ( 2 x - 2 x ^ { 2 } ) d x$

1. $\pi \int _ { 0 } ^ { 2 } ( \frac { y } { 2 } - \frac { y ^ { 2 } } { 2 } ) ^ { 2 } d x$

Sketch the graph and shade flag.

Sketch the axis of rotation. This will help you visualize the resultant 3D solid.

Choose a method:

$\text{Disks: }\pi \int_{x=a}^{x=b}(f(x))^{2}dx\text{ or }\pi \int_{y=a}^{y=b}(f(y))^{2}dy$

$\text{Washers: }\pi \int_{x=a}^{x=b}(f(x))^{2}-(g(x))^{2}dx\text{ or }\pi \int_{y=a}^{y=b}(f(y))^{2}-(g(y))^{2}dy$

$\text{Shell: }2\pi \int_{x=a}^{x=b}(x-k)(f(x))dx\text{ or }2\pi \int_{y=a}^{y=b}(y-k)(f(y))dy$

Determine the bounds of integration by solving a system of equations. You might write your bounds as $x$- or $y$- values, depending on the method you chose in the previous step.

Your integrand should should be written in terms of the same variable as the bounds.