Q38 / 90 · JEE Main 2022 (27 Jul Shift 2) · Mathematics

The area of the region enclosed by $y \leq 4 x^{2}, x^{2} \leq 9 y$ and $y \leq 4$ , is equal to

A
$\frac{40}{3}$
B
$\frac{56}{3}$
C
$\frac{112}{3}$
Correct answer
D
$\frac{80}{3}$

Solution

Plotting the required region
Enclosed area $= 2 \int_{0}^{4} (3 \sqrt{y} - \frac{\sqrt{y}}{2}) d y$
$= 2 \int_{0}^{4} \frac{5 \sqrt{y}}{2} d y$
$= 5 {[\frac{2}{3} y^{\frac{3}{2}}]}_{0}^{4}$
$= \frac{10}{3} (4)^{\frac{3}{2}} = \frac{80}{3}$

Hence, the correct answer is option: 'D'.

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