QJEEM12Maf3bb: JEE Main 2012 Mathematics - Class 12 JEEM12Maf3bb Solution: The area between the parabolas and and the straight line is :

Question

Verified Solution • 2025–26 Session moderate

The area between the parabolas

{x^2} = {y \over 4}

and

{x^2} = 9y

and the straight line

y=2

is :

Accepted Answer

Given curves

{x^2} = {y \over 4}

and

{x^2} = 9y

are the parabolas whose equations can be written as

y = 4{x^2}

and

y = {1 \over 9}{x^2}.

Also, given

y=2.

Now, shaded portion shows the required area which is symmetric.

AIEEE 2012 Mathematics - Area Under The Curves Question 132 English Explanation

\therefore

Area

= 2\int\limits_0^2 {\left( {\sqrt {9y} - \sqrt {{y \over 4}} } \right)} dy

Area

= 2\int\limits_0^2 {\left( {3\sqrt y - {{\sqrt y } \over 2}} \right)} dy

= 2\left[ {{2 \over 3} \times 3.{y^{{3 \over 2}}} - {1 \over 2} \times {2 \over 3}.{y^{{3 \over 2}}}} \right]_0^2

= 2\left[ {2{y^{{3 \over 2}}} - {1 \over 3}{y^{{3 \over 2}}}} \right] = \left. {2 \times {5 \over 3}{y^{{3 \over 2}}}} \right|_0^2

= 2.{5 \over 3}2\sqrt 2 = {{20\sqrt 2 } \over 3}

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