Question

The area (in sq. units) of the region enclosed by the curves $y=x^{2}-1$ and $y=1-x^{2}$ is equal to

• A. $\frac{4}{3}$
• B. $\frac{8}{3}$
• C. $\frac{16}{3}$
• D. $\frac{7}{2}$

Answer 1

Answer: (B)

$y=x^{2}-1$ and $y=1-x^{2}$

$A =\int\limits_{-1}^{1}\left(\left(1- x ^{2}\right)-\left( x ^{2}-1\right)\right) d x$
$A =\int\limits_{-1}^{1}\left(2-2 x ^{2}\right) dx =4 \int_{0}^{1}\left(1- x ^{2}\right) dx$
$A =4\left( x -\frac{ x ^{3}}{3}\right)_{0}^{1}=4\left(\frac{2}{3}\right)=\frac{8}{3}$