How do you find the average value of $f(x)=cosx$ as x varies between $[1,5]$ ?

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Answer 1 · 2016-11-23T21:47:28

$1/4(sin(5)-sin(1))approx-0.45010$

The average value of the function $f$ on the interval $[a,b]$ is

$1/(b-a)int_a^bf(x)dx$

With the given information this translates into

$1/(5-1)int_1^5cos(x)dx$

The antiderivative of $cos(x)$ is $sin(x)$ so

$=1/4[sin(x)]_1^5=1/4(sin(5)-sin(1))$

This is as simplified as we can get without using a calculator.

$1/4(sin(5)-sin(1))approx-0.45010$

How do you find the average value of f(x)=cosxf(x)=cosx as x varies between [1,5][1,5]?