CPM Homework Help : CALC Problem 8-70

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:

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

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

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

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

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

Step 1:

Sketch the graph and shade flag.

Step 2:

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

Step 3:

Choose a method:

Step 4:

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.

Step 5:

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