6.5 Average Value of a Function/2: Difference between revisions
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<math> | <math> | ||
\begin{align} | |||
f(x) = \sin{(4x)}\text{,}\quad [-\pi, \pi] | f(x) = \sin{(4x)}\text{,}\quad [-\pi, \pi] | ||
f_{avg} &= \frac{1}{\pi-(-\pi)}\int_{-\pi}^{\pi}\sin{(4x)}\,dx = \frac{1}{2\pi}\int_{-\pi}^{\pi}\sin{(4x)}\,dx | |||
f_{avg} = \frac{1}{\pi-(-\pi)}\int_{-\pi}^{\pi}\sin{(4x)}\,dx = \frac{1}{2\pi}\int_{-\pi}^{\pi}\sin{(4x)}\,dx | |||
&= \frac{1}{2\pi}\int_{-4\pi}^{4\pi}\sin{u}\frac{1}{4}\,du | |||
\end{align} | |||
<math> | <math> | ||
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New upper limit: <math>4\pi = 4(\pi)</math><br> | New upper limit: <math>4\pi = 4(\pi)</math><br> | ||
New lower limit: <math>-4\pi = 4(-\pi)</math> | New lower limit: <math>-4\pi = 4(-\pi)</math> | ||
Revision as of 17:53, 7 September 2022
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(x) = \sin{(4x)}\text{,}\quad [-\pi, \pi] f_{avg} &= \frac{1}{\pi-(-\pi)}\int_{-\pi}^{\pi}\sin{(4x)}\,dx = \frac{1}{2\pi}\int_{-\pi}^{\pi}\sin{(4x)}\,dx &= \frac{1}{2\pi}\int_{-4\pi}^{4\pi}\sin{u}\frac{1}{4}\,du \end{align} <math> \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)}