Question

The area of the region
$A=\left\{(x,y):0\le y\le x|x|+1\text{and}-1\le x\le 1\right\}$ in sq. units, is :

• A. $\frac{2}{3}$
• B. $\frac{1}{3}$
• C. $\frac{4}{3}$
• D. $2$

Answer 1

The correct option is D $2$

$A=\{\begin{array}{c}0\le y\le -x^2+1,-1<x<0\\ 0\le y\le x^2+1,0\le x<1\end{array}$


Area of shaded region, $A=\underset{-1}{\overset{0}{\int }}(-x^2+1)dx+\underset{0}{\overset{1}{\int }}(x^2+1)dx$
$={[-\frac{x^3}{3}+x]}_{-1}^0+{[\frac{x^3}{3}+x]}_0^1$
$=\left(0\right)-(\frac{1}{3}-1)+(\frac{1}{3}+1)-(0+0)$
$=2$ sq. units.