6.5 Average Value of a Function/17

1. Use the Average Value from a to b:

$f_{\text{avg}}={\frac {1}{b-a}}\int _{a}^{b}f(x)\,dx$

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 \frac{1}{12}\int_{0}^{12} 50 + 14\sin(\frac{\pi}{12}t)\,dt =\frac{1}{12}[50t-\frac{168}{\pi}\cos(\frac{\pi}{12}t)]\bigg|_{12}^{0}=\frac{1}{12}[(50)(12)-\frac{168}{\pi}\cos(\pi))(0-\frac{168}{\pi}\cos(0)]=\frac{1}{12}[600-\frac{168}{\pi}(-1)+\frac{168}{\pi}(1)] }

${\begin{aligned}u&={\frac {\pi }{12}}t\quad \quad \quad \quad \quad \int 14\sin(u){\frac {12}{\pi }}\,du\\\quad dt\cdot {\frac {du}{dt}}&=dt\quad \quad \quad \quad 14\cdot {\frac {12}{\pi }}\int \sin(u)\,du\\{\frac {12}{\pi }}du&=dt\quad \quad \quad \quad -{\frac {168}{\pi }}\cos(u)=-{\frac {168}{\pi }}\cos({\frac {\pi }{12}}t)\\\end{aligned}}$