Evaluating a piecewise linear integral as a triangle area

The graph of $f(x)$ consists of two line segments meeting at the point $(3,3)$: one rising from $(0,0)$ to $(3,3)$, and one falling from $(3,3)$ to $(6,0)$. Together with the x-axis, these segments enclose a triangle with base along the x-axis from $x=0$ to $x=6$ and a peak height of $3$ at $x=3$.

The triangle has base $6$ (from $x=0$ to $x=6$) and height $3$ (the maximum value of $f$). Since $f(x) \ge 0$ everywhere on $[0,6]$, the integral equals the area of this region: $\int_0^6 f(x)\,dx = \frac{1}{2}(6)(3) = 9.$

The peak height is three. The two line segments meet at the point where x equals three and f of x equals three, forming the apex of the triangle used to compute the area under the curve.