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Find the average value of the functions on the given interval.
Average value of
f(x)=x on
[5,13] :

Find the average value of the functions on the given interval. $\newline$ Average value of $f(x)=x$ on $[5,13]$

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Write equations of sine and cosine functions using properties

Full solution

Q. Find the average value of the functions on the given interval.

\newline

Average value of

f(x)=x

on

[5,13]

Identify Formula: Identify the formula for the average value of a function on an interval $[a, b]$ . $\newline$ The formula is: Average value = $\frac{1}{(b-a)} \int_{a}^{b} f(x) \, dx$ . $\newline$ Here, $f(x) = x$ , $a = 5$ , and $b = 13$ .
Calculate Integral: Calculate the integral of $f(x) = x$ from $5$ to $13$ . $\int_{5}^{13} x \, dx = \left[\frac{x^2}{2}\right]_{5}^{13} = \left(\frac{13^2}{2}\right) - \left(\frac{5^2}{2}\right) = \left(\frac{169}{2}\right) - \left(\frac{25}{2}\right) = \frac{144}{2} = 72$ .
Plug into Formula: Plug the integral result into the average value formula. $\newline$ Average value = $(1/(13-5)) \times 72 = (1/8) \times 72 = 9$ .

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