How would I find the average value of a function over a given interval?
By Emma Martinez •
How would I find the average value of this function? $$f(x)=\sin(nx),\ 0\le x\le \frac{\pi}{n}$$ Where n is a positive integer. I am not sure how I would solve this thank you.
$\endgroup$ 12 Answers
$\begingroup$Here, the average value $\overline{y}$ is \begin{align} \overline{y}&=\frac{1}{\pi/n}\int_0^{\pi/n}\sin(nx)\;dx\\[4pt] &=\frac{1}{\pi}\int_0^{ \pi/n}\sin (nx)\,ndx\\[4pt] &=\frac{1}{\pi}\int_0^{\pi}\sin t\;dt\qquad\qquad\text{being }t=nx\\ &=\color{blue}{\frac{2}{\pi}} \end{align}
$\endgroup$ 2 $\begingroup$The area under sine curve $ y = \sin x $ in the interval is 2 for interval $ 0<x<\pi $and average vale $2/\pi$, the way question is asked average value is independent of $n$.
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