Let $α$ be the area of the larger region bounded by the curve $y^{2} = 8 x$ and the lines $y = x a n d x = 2,$ which lies in the first quadrant. Then the value of $3 α$ is equal to______.

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answer is 22.

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Detailed Solution

$\begin{array}{l} A_{1} = \int_{0}^{2} (2 \sqrt{2} \sqrt{x} - x) dx = {(2 \sqrt{2} . \frac{2}{3} x^{3 / 2} - \frac{x^{2}}{2})}_{0}^{2} \\ = \frac{4 \sqrt{2}}{3} 2 \sqrt{2} - 2 = \frac{16}{3} - 2 = \frac{10}{3} \end{array}$
$\begin{array}{l} A_{2} = \int_{2}^{8} (2 \sqrt{2} \sqrt{x} - x) dx = {(2 \sqrt{2} . \frac{2}{3} x^{3 / 2} - \frac{x^{2}}{2})}_{2}^{8} \\ = \frac{4 \sqrt{2}}{3} 8 \sqrt{8} - 32 - \frac{10}{3} \\ = \frac{128}{3} - 32 - \frac{10}{3} = \frac{22}{3} = α \\ ∴ 3 α = 22 \end{array}$