Question

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

Answer 1

If `theta` goes from `theta_1` to `theta_2`, then the arc length is `sqrt{2}(e^{theta_2}-e^{theta_1})`.

Let us look at some details.
`L=int_{theta_1}^{theta_2}sqrt{r^2+({dr}/{d theta})^2}d theta`
since `r=e^{theta}` and `{dr}/{d theta}=e^{theta}`,
`=int_{theta_1}^{theta_2}sqrt{(e^{theta})^2+(e^{theta})^2}d theta`
by pulling `e^{theta}` out of the square-root,
`=int_{theta_1}^{theta_2}e^{theta}sqrt{2} d theta=sqrt{2}int_{theta_1}^{theta_2}e^{theta} d theta`
by evaluating the integral,
`=sqrt{2}[e^{theta}]_{theta_1}^{theta_2}=sqrt{2}(e^{theta_2}-e^{theta_1})`