Which integral gives the area enclosed by the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$?

Explanation

1. 01

   Solving for $y$ in the first quadrant: $y = \frac{b}{a}\sqrt{a^{2} - x^{2}}$, for $0 \le x \le a$.

2. 02

   Four equal quadrants give $\frac{4b}{a}\int_0^{a} \sqrt{a^{2} - x^{2}}\,dx = \pi ab$.