Rotate y = x^2, 0 \le x \le 2 around the y-axis. Volume by shells/cylindrical method:

Both give 8\pi Shells: 2\pi\int0^2 x \cdot x^2\,dx = 8\pi. Disks (in y): \pi\int0^4 (\sqrt y)^2\,dy = \pi \cdot 8 = 8\pi. (But for AP Calc AB the shell method is not required; disks are.)

\int0^2 2\pi x \cdot x^2\,dx = 8\pi

Volumes by Disk and Washer Methods

AP Calculus AB· difficulty 3/5

y = x^2

A
$4\pi$
B
$\int_0^4 \pi y\,dy = 8\pi$
C
$8\pi$
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D
$\int_0^2 2\pi x \cdot x^2\,dx = 8\pi$

Explanation

2\pi\int_0^2 x \cdot x^2\,dx = 8\pi

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