Area of the region containing all point ( x , y ) satisfying 0 ≤ y ≤ 4 − x 2 , y ≤ x 2 + x + 1 and y ≤ [ sin 2 4 x + cos 4 x ] is equal to

Question

Area of the region containing all point ( x , y ) satisfying 0 ≤ y ≤ 4 − x 2 ​ , y ≤ x 2 + x + 1 and y ≤ [ sin 2 4 x ​ + cos 4 x ​ ] is equal to

TestHub · Accepted Answer

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As $- 2 \leq X \leq 2$ , we have $1 \leq sin^{2} \frac{x}{4} + cos \frac{x}{4} < 2$ .

This implies $Y \leq 1$ .

The required area is given by:

$2 \int_{3}^{2} 4 - x^{2} d x + (3 - 1) + \int_{- 1}^{0} (x^{2} + x + 1) d x + 3$

This simplifies to $\frac{4 π - 1}{6} + 3$ sq. units.

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