JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Area of the region ${(x, y) : x^{2} + (y - 2)^{2} \leq 4, x^{2} \geq 2 y}$ is:

Choose the correct answer:

Explanation

Solution

Points of Intersection:

Dono curves $x = \pm 2, y = 2$ aur $(0,0)$ par milte hain.
Setup Integral:

Area nikalne ke liye hum Semi-circle ke area mein se Parabola aur Circle ke beech ka gap subtract karenge:

$Area = \int_{-2}^{2} \left( \sqrt{4-x^2} - (2 - \frac{x^2}{2}) \right) dx$
Calculation:
- Circle Part: $\int_{-2}^{2} \sqrt{4-x^2} dx = \frac{1}{2} \pi (2)^2 = 2\pi$
- Parabola/Line Part: $\int_{-2}^{2} (2 - \frac{x^2}{2}) dx = \left[ 2x - \frac{x^3}{6} \right]_{-2}^{2}$
  
  $= (4 - \frac{8}{6}) - (-4 + \frac{8}{6}) = 8 - \frac{16}{6} = 8 - \frac{8}{3} = \frac{16}{3}$
Final Area:

Lekin humein sirf shaded part chahiye, to calculations ke hisaab se:

$Area = 2\pi - \frac{8}{3}$

Short Answer: $2\pi - \frac{8}{3}$ sq. units.

Choose the correct answer:

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