Let be the area of the larger region bounded by the curve y 2 = 8 x and the lines y = x and x = 2 , which lies in the first quadrant. Then the value of 3 is equal to _____ . [2023]

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Solution

Q.
Let $α$ be the area of the larger region bounded by the curve $y^{2} = 8 x$ and the lines $y = x$ and $x = 2$ , which lies in the first quadrant. Then the value of $3 α$ is equal to _____ . [2023]

Ans.
(22)

We have $y^{2} = 8 x, y = x, x = 2$

The shaded region has the largest area. So,

$Area = \int_{2}^{8} (2 \sqrt{2} \sqrt{x} - x) d x$

$α = 2 \sqrt{2} \int_{2}^{8} x^{1 / 2} d x - \int_{2}^{8} x d x$

$= 2 \sqrt{2} \times \frac{2}{3} {[x^{3 / 2}]}_{2}^{8} - {[\frac{x^{2}}{2}]}_{2}^{8}$

$= \frac{4 \sqrt{2}}{3} [16 \sqrt{2} - 2 \sqrt{2}] - \frac{1}{2} [64 - 4] = \frac{112}{3} - 30 = \frac{22}{3}$

$Now 3 α = 22$

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