CalcRef

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.

Variables

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

Find the average value of f(x) = sin x on [0, π].
  1. f_avg = (1/π) ∫₀^π sin x dx = (1/π)[-cos x]₀^π = (1/π)(1+1) = 2/π

Answer: 2/π ≈ 0.6366

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