6.5 Average Value of a Function/2

${\begin{aligned}f(x)=\sin {(4x)}{\text{,}}\quad [-\pi ,\pi ]\\[2ex]f_{avg}&={\frac {1}{\pi -(-\pi )}}\int _{-\pi }^{\pi }\sin {(4x)}\,dx={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\sin {(4x)}\,dx\\[2ex]&={\frac {1}{2\pi }}\int _{-4\pi }^{4\pi }\sin {u}{\frac {1}{4}}\,du={\frac {1}{8pi}}\int _{-4\pi }^{4\pi }\sin(u)\,du\end{aligned}}$

${\begin{aligned}u&=4x\\[2ex]du&=4dx\\[2ex]{\frac {1}{4}}du&=dx\end{aligned}}$

New upper limit: $4\pi =4(\pi )$
New lower limit: $-4\pi =4(-\pi )$