The area of the region bounded by the curves $x$ $=y^2-2$ and $x=y$ is

• A

  $\frac{9}{4}$
• B

  $9$
• Correct answer
  C

  $\frac{9}{2}$
• D

  $\frac{9}{7}$

Solution

Given, $x=y^2-2$ and $x=y$. On solving, $x=y^2-2$ and $x=y$, we get $(-1,-1)$ and $(2,2)$. Area of the shaded region, $\begin{aligned} & A=\int_{-1}^2 y d y-\int_{-1}^2\left(y^2-2\right) d y \\ & =\left[\frac{y^2}{2}-\frac{y^3}{3}+2 y\right]_{-1}^2=\left(\frac{4}{2}-\frac{8}{3}+4\right)-\left(\frac{1}{2}+\frac{1}{3}-2\right) \\ & =\frac{10}{3}+\frac{7}{6}=\frac{27}{6}=\frac{9}{2} . \end{aligned}$

Hence, the correct answer is option: 'C'.