For $a > 0$, what is the area of the triangle bounded by the line $y = x + a$ and the coordinate axes?

Explanation

1. 01

   The line meets the axes at $(-a, 0)$ and $(0, a)$, and lies above the $x$-axis between them.

2. 02

   $\int_{-a}^{0} (x + a)\,dx = \left[\frac{x^{2}}{2} + ax\right]_{-a}^{0} = \frac{a^{2}}{2}$ square units.