Question

If the area bounded by the parabola $y=2-x^{2}$ and the line $y=-x$ is $\frac{k}{2}$ sq. units, then the value of $2k$ is equal to

• A. $9$
• B. $27$
• C. $18$
• D. $32$

Answer 1

Answer: (C)

Solving $y=2-x^{2}$ & $y=-x;$ we get $-x=2-x^{2}$
$\Rightarrow x=2,-1$
$\Rightarrow A\left(- 1,1\right)\& B\left(2 , - 2\right)$
Thus, the required area
$A=\int\limits _{- 1}^{2} \left(2 - x^{2}\right) - \left(- x\right) d x$
$=\left[2 x - \frac{x^{3}}{3} + \frac{x^{2}}{2}\right]_{- 1}^{2}$
$=\frac{9}{2}$
$\Rightarrow k=9$
$\therefore 2k=18$