QJEEM14Maac5d: JEE Main 2014 Mathematics - Class 12 JEEM14Maac5d Solution: The area of the region described by and is :

Question

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The area of the region described by

A = \left\{ {\left( {x,y} \right):{x^2} + {y^2} \le 1} \right.

and

\left. {{y^2} \le 1 - x} \right\}

is :

Accepted Answer

Given curves are

{x^2} + {y^2} = 1

and

{y^2} = 1 - x.

Intersection points are

x = 0,1

Area of shaded portion is the required area.

So, Required Area

=

Area of semi-circle

+

Area bounded by parabola

= {{\pi {r^2}} \over 2} + 2\int\limits_0^1 {\sqrt {1 - x} dx}

= {\pi \over 2} + 2\int\limits_0^1 {\sqrt {1 - x} \,dx}

( As radius of circle

=1

)

= \frac{π}{2} + 2 {[\frac{{(1 - x)}^{3 / 2}}{- 3 / 2}]}_{0}^{1}

= {\pi \over 2} - {4 \over 3}\left( { - 1} \right) = {\pi \over 2} + {4 \over 3}

Sq. unit

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