If the area of the region { ( x , y ) : 0 ? y ? min { 2 x , 6 x - x 2 } } is A , then 12A is equal to [2024]

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Solution

Q.
If the area of the region ${(x, y) : 0 \leq y \leq \min {2 x, 6 x - x^{2}}}$ is $A,$ then 12A is equal to ____________. [2024]

Ans.
(304)

Region ${(x, y) : 0 \leq y \leq \min (2 x, 6 x - x^{2})}$

$∴$ Let A be the area of the required region.

$A = 2 \int_{0}^{4} x d x + \int_{4}^{6} (6 x - x^{2}) d x$

$= {[x^{2}]}_{0}^{4} + {[3 x^{2} - \frac{x^{3}}{3}]}_{4}^{6}$

$= 16 + 3 (36 - 16) - \frac{1}{3} (216 - 64)$

$= 16 + 60 - \frac{152}{3} = \frac{228 - 152}{3} = \frac{76}{3} \Rightarrow A = \frac{76}{3} sq. units$

$∴$ $12 A = 12 \times \frac{76}{3} = 4 \times 76 = 304$

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