Lengths, Areas and Volumes

Volumes of Solids of Revolution

Volume of a function of $x$ rotated about the $x$-axis

$$ V = \int_a^b{\pi y^2}dx $$

Volume of a function of $y$ rotated about the $y$-axis

$$ V = \int_a^b{\pi x^2}dy $$

Polar Curves

Volume Enclosed by Polar Curve

Let $r = f(\theta)$ define the function $f(\theta)$ using polar coordinates. Then the area enclosed between two points $(\alpha, r_1)$ and $(\beta, r_2)$ on the curve is:

$$ A = \int_{\alpha}^{\beta} \frac{1}{2} r^2 d\theta $$

Arc Length for Polar Curves

$$ s = \int_{\alpha}^{\beta} \sqrt{r^2 + (\frac{dr}{d\theta})^2}d\theta $$

See https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM10-4-4.php for more good notes