Combine the sum of the two iterated integrals into a single integral by converting to polar coordinates. Evaluate the resulting iterated integral.

Question

Quizplus · Accepted Answer

We have the iterated integral $\int_0^1 \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}} (x^2+y^2) dx dy$. To convert to polar coordinates, we can use the transformations $x=r \cos 	heta$ and $y=r \sin 	heta$, along with the Jacobian $r$. The bounds of integration also change as follows:

$$0 \leq r \leq 1$$
$$-\frac{\pi}{2} \leq 	heta \leq \frac{\pi}{2}$$

The integral becomes $\int_{-\pi/2}^{\pi/2} \int_0^1 r^3 \cos^2 	heta + r^3 \sin^2 	heta \hspace{2mm} dr d	heta$. Simplifying the integrand gives $\int_{-\pi/2}^{\pi/2} \int_0^1 r^3 \hspace{2mm} dr d	heta$, which evaluates to $\frac{1}{4}$. Therefore, the answer is D.

Combine the Sum of the Two Iterated Integrals into a Single