What is the average value of the function&nbsp;f(x)=cos(8x)&nbsp;on the interval&nbsp;[0,π2]?

The average value of a function f over the interval (a,b) is given by ∫abf(x)dxb−a. Forexample, average velocity is given by ∫0τv(t)dtτSo in this case∫0π2f(x)dxπ2−0=∫0π2cos⁡(8x)dxπ2=[sin⁡(8x)8]0π2π2=[sin⁡(4π)8−sin⁡(0)8]π2=[08−08]π2=0

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High School
Calculus and Analysis
Calculus 1
Derivatives

Magdalena Warren

Answered question

2022-04-19

What is the average value of the function $f (x) = \cos (8 x)$ on the interval $[0, \frac{π}{2}]$ ?

Answer & Explanation

blogspainmarax6qi

Beginner2022-04-20Added 12 answers

The average value of a function f over the interval (a,b) is given by

\frac{\int_{a}^{b} f (x) d x}{b - a}

. For
example, average velocity is given by

\frac{\int_{0}^{τ} v (t) d t}{τ}

So in this case

\frac{\int_{0}^{\frac{π}{2}} f (x) d x}{\frac{π}{2} - 0} = \frac{\int_{0}^{\frac{π}{2}} \cos (8 x) d x}{\frac{π}{2}}

= \frac{{[\frac{\sin (8 x)}{8}]}_{0}^{\frac{π}{2}}}{\frac{π}{2}}

= \frac{[\frac{\sin (4 π)}{8} - \frac{\sin (0)}{8}]}{\frac{π}{2}}

= \frac{[\frac{0}{8} - \frac{0}{8}]}{\frac{π}{2}}

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