Volumes of Revolution

When rotating around an axis with no gap:

$V = \pi \int_a^b [R(x)]^2\,dx$

where $R(x)$ is the distance from the curve to the axis.

When there's a gap (two curves):

$V = \pi \int_a^b \left([R(x)]^2 - [r(x)]^2\right)\,dx$

$R$ = outer radius, $r$ = inner radius.

$R(x) = f(x)$. Integrate in $x$.

Either convert to $x = g(y)$ or use shell method (BC topic).

Adjust radii: distance from curve to the axis of rotation.

About $y = k$: $R = f(x) - k$ or $k - f(x)$ (whichever is positive).

Rotate $y = \sqrt{x}$ from $x = 0$ to $x = 4$ around x-axis.

$V = \pi \int_0^4 (\sqrt{x})^2\,dx = \pi \int_0^4 x\,dx = \pi[\frac{x^2}{2}]_0^4 = 8\pi$.