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NDA Math Conic Section

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Question : 11 of 34

Marks: +1, -0

What is the area of the region enclosed between the curve

y^{2}=2 x

and the straight line

y=x ?

$\frac{1}{2}$
$1$
$\frac{2}{3}$
$2$

Solution: 👈: Video Solution

Two given graphs are

y^{2}=2 x

and

y=x

.

Point of intersections are :

x^{2}=2 x\Rightarrow x^{2}-2 x=0

x(x-2)=0

x=0

and

2\& y=0

and 2

So, points of intersections are (0,0) and (2,2) . The graph are

The required area is the shaded portion

A=\left|\int\limits_{0}^{2} (y_{1}-y_{2})\cdot dx\right|

=\left|\int\limits_{0}^{2}(\sqrt{2}x-x)\cdot dx\right|

=\left[ \frac{2\sqrt{2}x^{3/2}}{3} - \frac{x^{2}}{2} \right]_{0}^{2}

=\frac{2\sqrt{2}}{3}(2)^{3/2}-2

=\frac{8}{3}-2=\frac{2}{3}

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