Evaluating a definite integral of |x| using triangle areas

The region under $f(x) = |x|$ from $-3$ to $3$ splits naturally at $x = 0$ into two right triangles: one for $x \in [-3, 0]$ and one for $x \in [0, 3]$. Each triangle has base length 3 (along the $x$-axis) and height 3 (the maximum value of $|x|$ reached at $x = -3$ and $x = 3$ respectively).

The area of each triangle is $\frac{1}{2}(3)(3) = \frac{9}{2}$. Since $f(x) \ge 0$ on the whole interval, both areas count as positive area under the curve, so $\int_{-3}^3 |x|\,dx = \frac{9}{2} + \frac{9}{2} = 9$.