# The area of the region

Question:

The area of the region

$\mathrm{A}=\{(x, y): 0 \leq y \leq x|x|+1$ and $-1 \leq x \leq 1\}$ in sq. units

is:

1. (1) $\frac{2}{3}$

2. (2) 2

3. (3) $\frac{4}{3}$

4. (4) $\frac{1}{3}$

Correct Option: , 2

Solution:

Given $A=\{(x, y): 0 \leq y \leq x|x|+1$ and $-1 \leq x \leq 1\}$

$\therefore$ Area of shaded region

$=\int_{-1}^{0}\left(-x^{2}+1\right) d x+\int_{0}^{1}\left(x^{2}+1\right) d x$

$=\left(-\frac{x^{3}}{3}+x\right)_{-1}^{0}+\left(\frac{x^{3}}{3}+x\right)_{0}^{1}$

$=0-\left(\frac{1}{3}-1\right)+\left(\frac{1}{3}+1\right)-(0+0)$

$=\frac{2}{3}+\frac{4}{3}=\frac{6}{3}=2 \quad$ square units