The area (in sq. units) of the region described by { ( x , y ) : y 2 ? 2 x , ? and y ? 4 x - 1 } is [2024]

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Solution

Q.
The area (in sq. units) of the region described by ${(x, y) : y^{2} \leq 2 x, and y \geq 4 x - 1}$ is [2024]

1 $\frac{11}{12}$

2 $\frac{11}{32}$

3 $\frac{9}{32}$

4 $\frac{8}{9}$

Ans.
(3)

$Required Area = \int_{- \frac{1}{2}}^{1} (\frac{y + 1}{4} - \frac{y^{2}}{2}) d y$

$= {[\frac{y^{2}}{8} + \frac{y}{4} - \frac{y^{3}}{6}]}_{- \frac{1}{2}}^{1}$

$= (\frac{1}{8} + \frac{1}{4} - \frac{1}{6}) - (\frac{1}{32} - \frac{1}{8} + \frac{1}{48}) = \frac{9}{32}$

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