Using integration, what is the area enclosed by the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1$?

Explanation

1. 01

   In the first quadrant $y = \frac{4}{5}\sqrt{25 - x^{2}}$; by symmetry the area is $4\int_0^{5} \frac{4}{5}\sqrt{25 - x^{2}}\,dx$.

2. 02

   $4 \times \frac{4}{5} \times \frac{25\pi}{4} = 20\pi$ square units, which is $\pi ab$ with $a = 5$, $b = 4$.