6.5 Average Value of a Function/2

$f(x)=\sin {(4x)}{\text{,}}\quad [-\pi ,\pi ]$

$x^{2}+y^{2}+z^{2}=1$

(1)

Eq. 1 $x^{2}+x+1=0$

$f_{avg}={\frac {1}{\pi -(-\pi )}}\int _{-\pi }^{\pi }\sin {(4x)}\,dx={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\sin {(4x)}\,dx$

(1)

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

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