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Wize University Calculus 1 Textbook > Applications of Integration

Average Value of a Function

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Average Value of a Function

If we could "level out" a function, making it flat, the height would represent the functions average.


Average Value

The average value of a function f(x)f(x) on an interval [a,b][a,b] is given by
Average(f(x))=1b−a∫abf(x)dx\boxed{\text{Average}(f(x))=\frac{1}{b-a}\int_{a}^{b}f(x)dx}
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Example: Average Value

Compute the average value of f(x)=3x2+6x+3\displaystyle f(x)=3x^2+6x+3 on the interval [2,3][2,3].

Average value =13−2∫23f(x)dx=∫23(3x2+6x+3)dx=[x3+3x2+3x]23=(27+27+9)−(8+12+6)=(54+9)−(26)=37\begin{aligned} \text{Average value } &= \dfrac{1}{3-2}\int_2^3f(x)dx\\ &= \int_2^3(3x^2+6x+3)dx\\ &= \left[x^3+3x^2+3x\right]_2^3\\ &= (27+27+9) - (8+12+6)\\ &= (54+9)-(26)\\ &= 37\\ \end{aligned}

Find the average value of the function f(x)=1x2f\left(x\right)=\frac{1}{x^2} over the interval [1, 3]\left[1,\ 3\right].

Extra Practice
Find the average value of the function f(θ)=sec⁡θtan⁡θf\left(\theta\right)=\sec\theta\tan\theta over the interval [0,π4]\left[0,\dfrac{\pi}{4}\right]
Average Value of Polynomial and Exponential Functions
Find the average value of the function g(x)=ax+xag\left(x\right)=a^x+x^a over the interval [2,3]\left[2,3\right] , where a ≠−1a\ \ne-1 is a constant.
Average Value of Polynomial and Exponential Functions
Find the average value of the function g(x)=ax+xag\left(x\right)=a^x+x^a over the interval [2,3]\left[2,3\right] , where a ≠−1a\ \ne-1 is a constant.
Average Value of a Function Problem
Find the average value of y=x2y=x^2 over [1,3][1, 3] .
Find the average value of the function f(x)=1x2f\left(x\right)=\frac{1}{x^2} over the interval [1, 3]\left[1,\ 3\right]
Average Value of a Function
Find the average value of f(x)=sin⁡xf(x)=\sin x on [0,π/2][0,\pi/2].
Average Value of a Function Problem
Find the average value of y=x2y=x^2 over [1,3][1, 3] .
Find the average value of the function f(θ)=sec⁡θtan⁡θf\left(\theta\right)=\sec\theta\tan\theta over the interval [0,π4]\left[0,\dfrac{\pi}{4}\right]
Find the average value of the function f(θ)=sec⁡θtan⁡θf\left(\theta\right)=\sec\theta\tan\theta over the interval [0,π4]\left[0,\dfrac{\pi}{4}\right]
Average Value of Polynomial and Exponential Functions
Find the average value of the function g(x)=ax+xag\left(x\right)=a^x+x^a over the interval [2,3]\left[2,3\right] , where a ≠−1a\ \ne-1 is a constant.