Evaluating a definite integral via triangle area

The graph of $f(x) = -|x|+4$ on $[-4,4]$ is a triangle with vertices at $(-4,0)$, $(4,0)$, and $(0,4)$. This triangle lies entirely above the $x$-axis (touching it only at the endpoints), so the signed area equals the ordinary area of the triangle.

The base of the triangle spans from $x=-4$ to $x=4$, so the base length is $8$, and the height (the peak value at $x=0$) is $4$. Using the area formula $\frac{1}{2}\cdot \text{base} \cdot \text{height}$, the area is $\frac{1}{2}(8)(4) = 16$. Therefore $\int_{-4}^{4}(-|x|+4)\,dx = 16$.