Use an iterated integral to find the area of the region shown in the figure below.

Question

Quizplus · Accepted Answer

We can use the iterated integral $\int_{-1}^{1} \int_{x^{2}}^{2-x^{2}} dy \, dx$ to find the area of the region. The limits of integration in the $x$ direction are from $-1$ to $1$, and for each value of $x$, the limits of integration in the $y$ direction are from the parabola $y=x^2$ to the line $y=2-x^2$. Therefore, integrating with respect to $y$ first, we get:

$$\int_{-1}^{1} \int_{x^{2}}^{2-x^{2}} dy \, dx = \int_{-1}^{1} (2-x^{2}-x^{2}) \, dx = \int_{-1}^{1} (2-2x^{2}) \, dx = \frac{8}{3}$$

Thus, the area of the region is $\frac{8}{3}$. The only choice that matches this value is $\boxed{	ext{(B)}}$.

Use an Iterated Integral to Find the Area of the Region