The area of the region enclosed by the parabola x 2 = y and the line y = x + 2 is:

Question

The area of the region enclosed by the parabola $x^{2} = y$ and the line $y = x + 2$ is:

Smart Achievers · Accepted Answer

(3)

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

Explanation:

We have, $x^{2} = y$ and $y = x + 2$

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

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

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

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

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

$x = - 1, 2$

$∴$ Required area of the shaded region,

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

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

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

$= 6 + \frac{3}{2} - \frac{9}{3}$

$= \frac{36 + 9 - 18}{6}$

$= \frac{27}{6}$

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

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