The area common to the curves $y^{2} = x$ and $x^{2} = y$ is

1

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\frac{2}{3}

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\frac{1}{3}

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None of these

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Solution

The correct option is C $\frac{1}{3}$
The given curves intersect at
$y = x^{2} = y^{4}$ : $y = 0$ or $y = 1$
$x = y^{2} = x^{4}$ : $x = 0$ or $x = 1$
$= \int_{x = 0}^{x = 1} (y_{1} - y_{2}) d x = \int_{x = 0}^{1} (\sqrt{x} - x^{2}) d x$
$= {⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} - \frac{x^{3}}{3} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦}_{0}^{1} = \frac{2}{3} - \frac{1}{3} = \frac{1}{3}$

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