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

$=\left(\frac{x^{2}}{2}+2 x-\frac{x^{3}}{3}\right)_{-1}^{2}$

$=\left(2+4-\frac{8}{3}\right)-\left(+\frac{1}{2}-2+\frac{1}{3}\right)=\frac{9}{2}$

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