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:
Show Hint
For area calculations involving curves, ensure to carefully analyze the region enclosed and utilize symmetry for simplified integration.
Step 1: Define the region bounded by curves. The curves are given as: - \( C_1 : |y| = 1 - x^2 \) - \( C_2 : x^2 + y^2 = 1 \) The enclosed area, denoted by \( \alpha \), is calculated as: \[\alpha = 4 \left[ \text{Area of the circle in the first quadrant} - \int_0^1 (1 - x^2) \, dx \right]\]
Step 2: Evaluate the necessary integrals. The area of the quarter circle is: \[ \text{Area} = \frac{\pi}{4} \] The integral is evaluated as: \[\int_0^1 (1 - x^2) \, dx = \left[ x - \frac{x^3}{3} \right]_0^1 = 1 - \frac{1}{3} = \frac{2}{3}\]
Step 3: Calculate the area difference. \[\alpha = 4 \left[ \frac{\pi}{4} - \frac{2}{3} \right]\] \[\alpha = \pi - \frac{8}{3}\]
Step 4: Determine \( 9\alpha \). \[9\alpha = 9\pi - 24\] Comparing this to \( 9\alpha = \beta\pi + \gamma \), we identify the coefficients: \[\beta = 9, \quad \gamma = 24\]
