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> | ||
| Line 21: | Line 23: | ||
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
![{\displaystyle {\begin{aligned}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{aligned}}<math>{\begin{aligned}u&=4x\\[2ex]du&=4dx\\[2ex]{\frac {1}{4}}du&=dx\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/1dd6636fa64f7deb3edb4913f9fb3804abfabfdf)
New upper limit:
New lower limit:
