If the area of the region is A, then 3A is equal to [2025]

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Solution

Q.
If the area of the region ${(x, y) : 1 + x^{2} \leq y \leq \min {x + 7, 11 - 3 x}}$ is A, then 3A is equal to [2025]

1 47

2 49

3 50

4 46

Ans.
(3)

$A = \int_{- 2}^{1} (x + 7 - x^{2} - 1) d x + \int_{1}^{2} (11 - 3 x - x^{2} - 1) d x$

$= {[\frac{x^{2}}{2} + 6 x - \frac{x^{3}}{3}]}_{- 2}^{1} + {[10 x - \frac{3 x^{2}}{2} - \frac{x^{3}}{3}]}_{1}^{2}$

$= \frac{50}{3} \Rightarrow 3 A = 50$ .

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