Finding a function's average value over a particular interval
What is the average value of a function?
In the same way that we can find the average of set of numbers, we can also find the average value of a function over a specific interval.
The formula we use to find the average value of a function ???f(x)??? over the interval ???[a,b]??? is
???f_{avg}=\frac{1}{b-a}\int^b_af(x)\ dx???
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Think about the average value of a function as the average height the function attains above the ???x???-axis. If the function were ???y=3???, then the height of the function is always ???3??? everywhere, so the average height of the function would also be ???3???. When the function gets more complicated, we can use the average value formula to find its average height on ???[a,b]???.
How to calculate a function’s average value over a particular interval
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Using the average value integration formula
Example
Calculate the average value of the function over the interval.
???f(x)=x^3-2x^2+e^{2x}???
on ???[3,7]???
We’ll use the formula for average value
???f_{avg}=\frac{1}{b-a}\int^b_af(x)\ dx???
and get
???f_{avg}=\frac{1}{7-3}\int^7_3x^3-2x^2+e^{2x}\ dx???
???f_{avg}=\frac{1}{4}\int^7_3x^3-2x^2+e^{2x}\ dx???
Think about the average value of a function as the average height the function attains above the x-axis.
Next we can break the integral apart by term.
???f_{avg}=\frac{1}{4}\int^7_3x^3\ dx+\frac{1}{4}\int^7_3-2x^2\ dx+\frac{1}{4}\int^7_3e^{2x}\ dx???
???f_{avg}=\frac{1}{4}\int^7_3x^3\ dx-\frac{2}{4}\int^7_3x^2\ dx+\frac{1}{4}\int^7_3e^{2x}\ dx???
Integrate.
???f_{avg}=\frac14\left(\frac{x^4}{4}\right)\bigg|^7_3-\frac24\left(\frac{x^3}{3}\right)\bigg|^7_3+\frac14\left(\frac{e^{2x}}{2}\right)\bigg|^7_3???
???f_{avg}=\frac{x^4}{16}-\frac{x^3}{6}+\frac{e^{2x}}{8}\bigg|^7_3???
Now we can evaluate on the interval.
???f_{avg}=\left[\frac{(7)^4}{16}-\frac{(7)^3}{6}+\frac{e^{2(7)}}{8}\right]-\left[\frac{(3)^4}{16}-\frac{(3)^3}{6}+\frac{e^{2(3)}}{8}\right]???
???f_{avg}=150,367???
The average value of the function ???f(x)=x^3-2x^2+e^{2x}??? over the interval ???[3,7]??? is ???150,367???.
