The area bounded by y = 2 - x2 and x + y = 0 is:

Question

The area bounded by $y = 2 - x^{2}$ and $x + y = 0$ is:

Smart Achievers · Accepted Answer

(2)

Ans. $\frac{9}{2}$ sq. units

Explanation :

$y = 2 - x^{2}$ and $x + y = 0$

$\Rightarrow$ $x^{2} - x - 2 = 0$

$\Rightarrow$ $x^{2} - 2 x + x - 2 = 0$

$\Rightarrow$ $(x - 2) (x + 1) = 0$

$\Rightarrow$ $x = 2, - 1$

$x = 2$ then $y = - 2$

$x = - 1$ then $y = 1$

Required area,

$A = \int_{- 1}^{2} (2 - x^{2}) d x - \int_{- 1}^{2} (- x) d x$

$= {[2 x - \frac{x^{3}}{3} + \frac{x^{2}}{2}]}_{- 1}^{2}$

$= 4 - \frac{8}{3} + \frac{4}{2} + 2 + \frac{1}{3} + \frac{1}{2}$

$= 6 + 2 - 3 - \frac{1}{2} = \frac{9}{2} sq. units.$

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