Calculus / Unit 9: Applications of Integrals
9.3 Average Value of a Function
Notes
The average value of $f$ on $[a, b]$ is $\displaystyle f_{\text{avg}} = \frac{1}{b - a}\int_a^b f(x)\,dx$.
- It is the height of a rectangle over $[a, b]$ with the same area as the region under $f$.
- Example: $f(x) = x^2$ on $[0, 3]$. $\displaystyle \int_0^3 x^2\,dx = 9$, so $\displaystyle f_{\text{avg}} = \frac{9}{3} = 3$.
- Do not forget to divide by the length of the interval.
Practice
Find the average value of $f(x) = x^2$ on $[0, 3]$.
$\displaystyle \frac{1}{3}\int_0^3 x^2\,dx = \frac{1}{3}(9) = 3$.
Find the average value of $f(x) = \sin x$ on $[0, \pi]$.
$\displaystyle \frac{1}{\pi}\int_0^\pi \sin x\,dx = \frac{2}{\pi}$.
Find the average value of $f(x) = 2x + 1$ on $[1, 4]$.
$\displaystyle \frac{1}{3}\left[x^2 + x\right]_1^4 = \frac{20 - 2}{3} = 6$.
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