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Q.

If the area bounded by the curve $2 y^{2} = 3 x$ , lines $x + y = 3, y \geq 0$ and outside the circle
${(x - 3)}^{2} + y^{2} = 2$ is

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a

$\frac{21}{2} - \frac{π}{4}$

b

$\frac{21}{4} - \frac{π}{4}$

c

$\frac{21 - π}{2}$

d

$\frac{21}{8} - \frac{π}{4}$

answer is A.

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Detailed Solution

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$A = \int_{0}^{3 / 2} ((3 - y) - \frac{2 y^{2}}{3}) d y - \frac{2 π}{8} = {(3 y - \frac{y^{2}}{2} - \frac{2 y^{3}}{9})}^{3 / 2} - \frac{2 π}{8} = \frac{9}{2} - \frac{9}{8} - \frac{2 \times 27}{8 \times 9} - \frac{2 π}{8} = \frac{36 - 9 - 6}{8} - \frac{π}{4} = \frac{21}{8} - \frac{π}{4}$

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