APFIVE Volume with Rectangular Cross Sections
The region enclosed by the graphs of $$ y = x^2 $$ and $$ y = 2x $$ is the base of a solid. For this solid, cross sections taken perpendicular to the $$y$$-axis are rectangles whose height is 3 times the length of their base in the $$xy$$-plane. Which of the following integrals gives the volume of the solid?
A
$$ 3 \int_0^4 \left( \sqrt{y} - \frac{y}{2} \right)^2 \, dy $$
B
$$ 3 \int_0^4 \left( \sqrt{y} + \frac{y}{2} \right)^2 \, dy $$
C
$$ 3 \int_0^2 (2x - x^2)^2 \, dx $$
D
$$ 3 \int_0^2 (2x + x^2)^2 \, dx $$
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