If the area of the region \ (x,y): 4 - x 2 le y le x 2, \, y le 4, \, x ge 0\ is 80√(2)α - β , where α, β in mathbb N , then α + β is equal to

Mathematics · Area under the Curves

JEE Main 2025 — 2 April, Morning Shift — Question 43

If the area of the region $\{(x,y): |4 - x^2| \le y \le x^2, \, y \le 4, \, x \ge 0\}$ is $80\sqrt{2}\alpha - \beta$ , where $\alpha, \beta \in \mathbb{N}$ , then $\alpha + \beta$ is equal to

Answer: 22

Numerical answer — enter this value.

Step-by-step solution

Area= $\int_{\sqrt{2}}^{2}\left(x^{2}-\left(4-x^{2}\right)\right) d x+(2 \sqrt{2}-2) \times 4-\int_{2}^{2 \sqrt{2}}\left(x^{2}-4\right) d x$

$=\left[\frac{2 x^{3}}{3}-4 x\right]_{\sqrt{2}}^{2}+8 \sqrt{2}-8-\left[\frac{x^{3}}{3}-4 x\right]_{2}^{2 \sqrt{2}}$

$=\frac{40 \sqrt{2}}{3}-16$

$\Rightarrow \alpha=6, \beta=16 \Rightarrow \alpha+\beta=22$

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 2 April, Morning Shift
Subject: Mathematics
Chapter: Area under the Curves
Topic: Area under the Curves