# The area (in sq. units) of the region

Question:

The area (in sq. units) of the region $A=\left\{(x, y): x^{2} \leq y \leq x+2\right\}$ is:

1. (1) $\frac{10}{3}$

2. (2) $\frac{9}{2}$

3. (3) $\frac{31}{6}$

4. (4) $\frac{13}{6}$

Correct Option: , 2

Solution:

Required area is equal to the area under the curves $y \geq x^{2}$ and $y$\therefore$requried area$\int_{-1}^{2}\left((x+2)-x^{2}\right) d x=\left(\frac{x^{2}}{2}+2 x-\frac{x^{3}}{3}\right)_{-1}^{2}=\left(2+4-\frac{8}{3}\right)-\left(+\frac{1}{2}-2+\frac{1}{3}\right)=\frac{9}{2}\$