Source: Extra Practice
Find the area of the circle x² + y² = a² using the method of integration.
Options
Option A is correct
πa² sq. units
Option B
2πa² sq. units
Option C
πa²/2 sq. units
Option D
4πa² sq. units
Explanation
Area = 4 * ∫[0 to a] √(a² - x²) dx (considering the first quadrant). Using the standard formula ∫√(a² - x²) dx = (x/2)√(a² - x²) + (a²/2)sin⁻¹(x/a), we evaluate from 0 to a. At x=a, it's (a²/2)(π/2) = πa²/4. Multiplying by 4 gives πa². Options B, C, and D are incorrect multiples of the area of a circle.