Evaluate the iterated integral by converting to polar coordinates.

Question

Evaluate the iterated integral  by converting to polar coordinates.

Quizplus · Accepted Answer

Converting to polar coordinates, we have:
$$\int_{-1}^{1}\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}e^{x^2+y^2}dydx=\int_{0}^{\pi}\int_{0}^{1}e^{r^2}rdrd	heta$$
The limits of integration for $r$ are from $0$ to $1$ because the region of integration is a unit disk centered at the origin. The integral of $r$ is just $\frac{1}{2}$, and the integral of $e^{r^2}$ is well-known to be $\frac{\sqrt{\pi}}{2}$, so the answer is:
$$\frac{\sqrt{\pi}}{2}\cdot\frac{1}{2}=\frac{\sqrt{\pi}}{4}$$
Therefore, the correct choice is $	extbf{(E)}$

Evaluate the Iterated Integral by Converting to Polar Coordinates