Find the line integral of $x^2+y^2$ over the polar curve $r=e^{\theta}$

Question

Find the line integral of $x^2+y^2$ over the polar curve $r=e^{\theta}$

Not really sure on how to find the curve on terms of a parameter in order to evaluate the integral

Answer 1

Note that $x^2 + y^2 = r^2$, so the integral you're trying to find is $\int r^2\, ds$, where $ds$ is the arc-length form $$ds = \sqrt{r^2 + \left( \frac{dr}{d\theta}\right)^2}\, d\theta$$ In this case $r = \frac{dr}{d\theta} = e^{\theta}$, so plugging in gives $$\int r^2 ds = \int (e^\theta)^2 \sqrt{2 (e^\theta)^2}\, d\theta = \int \sqrt{2} e^{3\theta}\, d\theta. $$