The area (in sq. units) of the region described by \ ( x , y ): y 2 leq 2 x . , and . y geq 4 x -1 \ is

Mathematics · Area under the Curves

JEE Main 2024 — 4 April, Shift 2 — Question 9

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

Option A:
$\frac{11}{32}$
Option B:
$\frac{8}{9}$
Option C:
$\frac{11}{12}$
Option D:
$\frac{9}{32}$
Correct

Answer: D

Step-by-step solution

Shaded area $=\int_{-\frac{1}{2}}^{1}\left(\mathrm{x}_{\text {Right }}-\mathrm{x}_{\text {Left }}\right) \mathrm{dy}$

$y^{2}=2 x$

$y=4 x-1$

$y=1, y=-\frac{1}{2}$

shaded area $=\int_{-\frac{1}{2}}^{1}\left(\frac{y+1}{4}-\frac{y^{2}}{2}\right) d y$

$=\left(\frac{1}{4}\left(\frac{y^{2}}{2}+y\right)-\frac{y^{3}}{6}\right)_{-\frac{1}{2}}^{1}=\frac{9}{32}$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 4 April, Shift 2
Subject: Mathematics
Chapter: Area under the Curves
Topic: Area under the Curves