6.5 Average Value of a Function/17

1. Use the Average Value from a to b:

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

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

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