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# Choose the correct answer. Area lying between the curves y^2 = 4x and y = 2x is:

Q : 7         Area lying between the curves $\dpi{100} \small y^2=4x$ and $\dpi{100} \small y=2x$ is

(A)   $\dpi{100} \small \frac{2}{3}$            (B)  $\dpi{100} \small \frac{1}{3}$            (C)  $\dpi{100} \small \frac{1}{4}$            (D)  $\dpi{100} \small \frac{3}{4}$

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The area lying between the curve,  $\dpi{100} \small y^2=4x$ and $\dpi{100} \small y=2x$ is represented by the shaded area OBAO as

The points of intersection of these curves are $O(0,0)$ and $A (1,2)$.

So, we draw AC perpendicular to x-axis such that the coordinates of C are (1,0).

Therefore the Area OBAO =  $Area(\triangle OCA) -Area (OCABO)$

$=2\left [ \frac{x^2}{2} \right ]_0^1 - 2\left [ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right ]_0^1$

$=\left | 1-\frac{4}{3} \right | = \left | -\frac{1}{3} \right | = \frac{1}{3} units.$

Thus the correct answer is B.

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