Evaluate the iterated integral by converting to polar coordinates.Round the answer to two decimal places.

Question

Quizplus · Accepted Answer

The integral is: 
$$\int_{-2}^{2}\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}\frac{1}{\sqrt{x^2+y^2+1}}dydx$$
In polar coordinates, $x=r\cos(	heta)$ and $y=r\sin(	heta)$, so we have:
$$\int_{0}^{2\pi}\int_{0}^{2}\frac{r}{\sqrt{r^2+1}}drd	heta$$
Evaluating the integral:
$$\int_{0}^{2\pi}\int_{0}^{2}\frac{r}{\sqrt{r^2+1}}drd	heta=\int_{0}^{2\pi}\bigg[\sqrt{r^2+1}\bigg]_0^2d	heta=4\pi\sqrt{2}-4\pi\approx\boxed{2.50}$$
Therefore, the best choice is (C).

Evaluate the Iterated Integral by Converting to Polar Coordinates