${\int }_0^{1/2} |\sin 4\pi x|dx=$

$\begin{array}{l}:{\int }_0^{1/2} |\sin (4\pi x\left)\right|dx={\int }_0^{1/4} \sin \left(4\pi x\right)dx+{\int }_{1/4}^{1/2} [-\sin (4\pi x\left)\right]dx\\ ={\left[-\frac{\cos \left(4\pi x\right)}{4\pi }\right]}_0^{1/4}+{\left[\frac{\cos \left(4\pi x\right)}{4\pi }\right]}_{1/4}^{1/2}\\ =\frac{1}{4\pi }+\frac{1}{4\pi }+\frac{1}{4\pi }+\frac{1}{4\pi }=\frac{1}{\pi }\end{array}$