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

A) $\frac { 1024 } { 3 }$
B) $\frac { 16 } { 3 }$
C) $\frac { 125 } { 3 }$
D) $\frac { 17 } { 3 }$
E) $\frac { 128 } { 3 }$

Question

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

A) $\frac { 1024 } { 3 }$ 
B) $\frac { 16 } { 3 }$ 
C) $\frac { 125 } { 3 }$ 
D) $\frac { 17 } { 3 }$ 
E) $\frac { 128 } { 3 }$

Quizplus · Accepted Answer

The region can be expressed as the iterated integral $\int_{-1}^{1}\int_{x^2}^{1} dydx$. Evaluating this integral gives us $\int_{-1}^{1} (1-x^2) dx = \frac{2}{3} [x-\frac{x^3}{3}]_{-1}^{1} = \frac{2}{3}[\frac{2}{3}] = \frac{4}{9}$. However, we are looking for the area of the region which is the absolute value of the integral, giving us $\frac{8}{9}$. Finally, multiplying by 2 because the region is symmetric about the y-axis, we get $\frac{16}{9}$. Simplifying, we get the answer to be $\boxed{	extbf{(B) }\frac{16}{3}}$.

Use an Iterated Integral to Find the Area of the Region