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] }

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 \begin{align} 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{align} }

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

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)}