6.5 Average Value of a Function/17

1. Use the Average Value from a to b:

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_{\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 + \underbrace{14\sin\left(\frac{\pi}{12}t\right)}_{ \begin{aligned} u &= \frac{\pi}{12}t\\dt\cdot\frac{du}{dt} &= dt\\ \frac{12}{\pi}du &= dt \\ integrate for\, 14\sin(u)\frac{12}{\pi}\\ \int14\sin(u)\frac{12}{\pi}\,du 14\cdot\frac{12}{\pi}\int\sin(u)\,du \\ -\frac{168}{\pi}\cos(u) \\ -\frac{168}{\pi}\cos(\frac{\pi}{12}t) \end{aligned}} \,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)] }

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ={\frac {1}{12}}[600-{\frac {168}{\pi }}(-1)+{\frac {168}{\pi }}(1)]={\frac {1}{12}}[600+{\frac {168}{\pi }}+{\frac {168}{\pi }}]={\frac {1}{12}}[600+{\frac {336}{\pi }}]=50+{\frac {336}{12\pi }}=50+{\frac {28}{\pi }}=59}