High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize AP Calculus (AB) Textbook > Applications of Integration

Average Value of a Function

Non-Linear DEs [Coming Soon!]Previous Section
Distance and DisplacementNext Section
Popular Courses
Find My Course
0:00 / 0:00

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}
0:00 / 0:00

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].