Question

How do you find the length of the polar curve `r=theta` ?

Answer 1

If `theta` goes from `theta_1` to `theta_2`, then the arc length of can be found by
`L=int_{theta_1}^{theta_2}sqrt{theta^2+1}d theta`.

Since `r=theta`, which gives `{dr}/{d theta}=1`, we have
#L=int_{theta_1}^{theta_2}sqrt{r^2+({dr}/{d theta})^2}d theta =int_{theta_1}^{theta_2}sqrt{theta^2+1}d theta#