If the area of the region bounded by the curves y 2 - 2 y = - x and x + y = 0 is A, then 8A is equal to _________ . [2023]

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Solution

Q.
If the area of the region bounded by the curves $y^{2} - 2 y = - x$ and $x + y = 0$ is A, then 8A is equal to _________ . [2023]

Ans.
(36)

Required area $= \int_{0}^{3} [(2 y - y^{2}) - (- y)] d y$

$= \int_{0}^{3} (3 y - y^{2}) d y$

$= {[\frac{3 y^{2}}{2} - \frac{y^{3}}{3}]}_{0}^{3}$

$= \frac{27}{2} - \frac{27}{3} = \frac{9}{2} sq. units$

$∴$ $8 A = 8 \times \frac{9}{2} = 36$

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