$\begin{aligned} \int_0^4|2 x-5| \mathrm{d} x & =\int_0^{\frac{5}{2}}(5-2 x) \mathrm{d} x+\int_{\frac{5}{2}}^4(2 x-5) \mathrm{d} x \\ & =\left[5 x-x^2\right]_0^{\frac{5}{2}}+\left[x^2-5 x\right]_{\frac{5}{2}}^4 \\ & =\frac{25}{4}+\frac{9}{4} \\ & =\frac{34}{4} \\ & =\frac{17}{2}\end{aligned}$