6.5 Average Value of a Function/2

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(x) = \sin{(4x)}\text{,}\quad [-\pi, \pi] }

${\begin{aligned}f_{avg}&={\frac {1}{\pi -(-\pi )}}\int _{-\pi }^{\pi }\sin {(4x)}\,dx={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\sin {(4x)}\,dx\\[2ex]&={\frac {1}{2\pi }}\int _{-4\pi }^{4\pi }\sin {(u)}\left({\frac {1}{4}}\,du\right)={\frac {1}{8\pi }}\int _{-4\pi }^{4\pi }\sin(u)\,du\\[2ex]&=-{\frac {1}{8\pi }}\cos(u){\bigg |}_{-4\pi }^{4\pi }\\[2ex]&=\left[-{\frac {1}{8\pi }}\cos(4\pi )\right]-\left[-{\frac {1}{8\pi }}\cos(-4\pi )\right]=\left[-{\frac {1}{8\pi }}(1)\right]+\left[{\frac {1}{8\pi }}(1)\right]\\[2ex]&=0\end{aligned}}$

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

New upper limit: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 4\pi = 4(\pi)}
New lower limit: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -4\pi = 4(-\pi)}