$\underset{0}{\overset{4}{\int }}$ |x| + |x - 2| + |x - 4| dx
= $\underset{0}{\overset{2}{\int }}$ f (x) dx + $\underset{2}{\overset{4}{\int }}$ f (x) dx
= $\underset{0}{\overset{2}{\int }}$ x - x + 2 - x + 4 dx + $\underset{2}{\overset{4}{\int }}$ x + x - 2 - x + 4 dx
= $\underset{0}{\overset{2}{\int }}$ 6 - x dx + $\underset{2}{\overset{4}{\int }}$ x + 2 dx
= ${\left|6\mathit{x}-\frac{{\mathit{x}}^2}{2}\right|}_0^2$ + ${\left|\frac{{\mathit{x}}^2}{2}+2\mathit{x}\right|}_0^4$
= 12 - 2 + 8 + 8 - 2 - 4 = 20