What fraction of the area enclosed by the circle $x^{2} + y^{2} = 25$ does the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$ enclose?

Explanation

1. 01

   The ellipse has $a = 5$, $b = 3$: it encloses $\frac{4 \times 3}{5}\int_0^{5} \sqrt{25 - x^{2}}\,dx = 15\pi$; the circle encloses $25\pi$.

2. 02

   $\frac{15\pi}{25\pi} = \frac{3}{5}$, which is $\frac{b}{a}$: each vertical strip of the ellipse is $\frac{3}{5}$ of the circle's strip.