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