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

JEE Main · 2015 · 04 Apr 2015 · Offline · Area Under Curves

The area (in sq. units) of the region described by

{(x, y) : y^{2} \leq 2 x

and

y \geq 4 x - 1}

is :

Answer: (C)

The region lies inside the parabola

y^{2} = 2 x

and above the line

y = 4 x - 1

. They cross where

(4 x - 1)^{2} = 2 x

, i.e.

16 x^{2} - 10 x + 1 = 0

, at

x = \frac{1}{8}, \frac{1}{2}

with

y = - \frac{1}{2}

and

y = 1

. Slicing horizontally, each strip runs from the parabola

x = \frac{y ^{2}}{2}

on the left to the line

x = \frac{y + 1}{4}

on the right, so the area is

\int_{- 1/2}^{1} (\frac{y + 1}{4} - \frac{y ^{2}}{2}) d y = [\frac{y ^{2}}{8} + \frac{y}{4} - \frac{y ^{3}}{6}]_{- 1/2}^{1} = \frac{9}{32}

JEE Main · 2015 · 04 Apr 2015 · Offline · Single correct option