Twice the area (in sq. units) of the region and is

Question

Twice the area (in sq. units) of the region ${(x, y) : x \geq 0, x + y \leq 3, x^{2} \leq 4 y$ and $y \leq 1 + \sqrt{x}}$ is

Smart Achievers · Accepted Answer

(5)

$Let I = \int_{0}^{1} (1 + \sqrt{x}) d x + \int_{1}^{2} (3 - x) d x - \int_{0}^{2} \frac{x^{2}}{4} d x$

$= {[x + \frac{x^{3 / 2}}{3 / 2}]}_{0}^{1} + {[3 x - \frac{x^{2}}{2}]}_{1}^{2} -$ ${[\frac{x^{3}}{12}]}_{0}^{2}$

$= (1 + \frac{2}{3}) + (6 - 2 - 3 + \frac{1}{2}) - (\frac{8}{12})$

$= \frac{5}{3} + \frac{3}{2} - \frac{2}{3}$

$= 1 + \frac{3}{2}$

$= \frac{5}{2} sq. unit$

$∴$ $2 I = 5$

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