Signed area of a triangle pair for f(x) = 2x

On $[-3, 0]$, the graph is a triangle below the $x$-axis with base $3$ (from $x=-3$ to $x=0$) and height $|2(-3)| = 6$. Its area is $\frac{1}{2}(3)(6) = 9$, but since this region lies below the axis, it contributes $-9$ to the integral.

On $[0, 3]$, the graph is a triangle above the $x$-axis with base $3$ and height $2(3) = 6$. Its area is $\frac{1}{2}(3)(6) = 9$, and since it lies above the axis, it contributes $+9$.

Adding the two signed areas gives $-9 + 9 = 0$. Therefore $\int_{-3}^{3} 2x\, dx = 0$, since the positive and negative triangular regions exactly cancel.