For $a, b > 0$, what is the area of the smaller region bounded by the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and the line $\frac{x}{a} + \frac{y}{b} = 1$?

Explanation

1. 01

   The line joins $(a, 0)$ and $(0, b)$, both on the ellipse, so the smaller region lies in the first quadrant between the arc and the chord.

2. 02

   $\int_0^{a} \left[\frac{b}{a}\sqrt{a^{2} - x^{2}} - b\left(1 - \frac{x}{a}\right)\right]dx = \frac{\pi ab}{4} - \frac{ab}{2}$.

3. 03

   $= \frac{(\pi - 2)ab}{4}$ square units.