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Integrals · Special Integrals

Average Value of a Function

favg=1b−a∫abf(x) dxf_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x) \, dx

The average value of f on [a, b] is the integral divided by the interval length.

  • 2 variables
  • 2 worked examples
  • 19 in Integrals

Variables

Variables used in the Average Value of a Function formula
SymbolNameUnit
aLower bound -
bUpper bound -

Worked examples

Find the average value of f(x) = x² on [0, 3].
  1. f_avg = (1/3) ∫₀³ x² dx = (1/3)[x³/3]₀³ = (1/3)(9) = 3

Answer: 3

1 more worked examplePremium

Find the average value of f(x) = sin x on [0, π].

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Common questions

What is the Average Value of a Function?

The average value of f on [a, b] is the integral divided by the interval length. It is one of the integrals formulas in the CalcRef reference.

What do the symbols in the Average Value of a Function mean?

In the Average Value of a Function, a is the lower bound; b is the upper bound.

How do you use the Average Value of a Function?

Worked example. Find the average value of f(x) = x² on [0, 3]. f_avg = (1/3) ∫₀³ x² dx = (1/3)[x³/3]₀³ = (1/3)(9) = 3. Answer: 3.

Practice

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