The area included between the parabola $y^{2} = 4 x$ and the line $y = 2 x$ is ____ sq. units

\frac{1}{3}

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\frac{2}{3}

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\frac{4}{3}

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\frac{5}{3}

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Solution

The correct option is B $\frac{1}{3}$
Area $= \int_{0}^{1} [2 \sqrt{x} - 2 x]$

$= 2 {⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} - \frac{x^{2}}{2} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦}_{0}^{1}$

$= 2 (\frac{2}{3} - \frac{1}{2}) \times 1$

$= \frac{4}{3} - 1$

$\frac{1}{3}$ sq. units

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