Question

Let the area enclosed between the curves \( |y| = 1 - x^2 \) and \( x^2 + y^2 = 1 \) be \( \alpha \). If \( 9\alpha = \beta\pi + \gamma \); \( \beta, \gamma \) are integers, then the value of \( |\beta - \gamma| \) equals:

• A. 27
• B. 18
• C. 15
• D. 33

Hint:

For area calculations involving curves, ensure to carefully analyze the region enclosed and utilize symmetry for simplified integration.

Answer 1

The Correct Option is D

Step 1: Identifying the area enclosed between curves. Given curves: - \( C_1 : |y| = 1 - x^2 \) - \( C_2 : x^2 + y^2 = 1 \) The area enclosed between these curves is calculated by: \[ \alpha = 4 \left[ \text{Area of circle in 1st quadrant} - \int_0^1 (1 - x^2) \, dx \right] \]

Step 2: Compute the required integrals. Area of the quarter circle is: \[ \text{Area} = \frac{\pi}{4} \] Now, evaluating the integral: \[ \int_0^1 (1 - x^2) \, dx = \left[ x - \frac{x^3}{3} \right]_0^1 = 1 - \frac{1}{3} = \frac{2}{3} \]

Step 3: Finding the area difference. \[ \alpha = 4 \left[ \frac{\pi}{4} - \frac{2}{3} \right] \] \[ \alpha = \pi - \frac{8}{3} \]

Step 4: Calculating \( 9\alpha \). \[ 9\alpha = 9\pi - 24 \] Here, comparing with \( 9\alpha = \beta\pi + \gamma \), we get: \[\beta = 9, \quad \gamma = 24\]

Step 5: Finding \( |\beta - \gamma| \) \[ |\beta - \gamma| = |9 - 24| = 33 \]