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Calculus basics

Average value of a function

The height of the rectangle on [a, b] with the same area as the region under the curve.

f¯=1b−a∫abf(x)dx

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−1−0.50.511.522.533.542468101214
  • Integral from 0 to 3 ≈ 9 (shaded; parts below the axis count as negative).
  • Average value 3 (dashed): that rectangle has the same area.

What the symbols mean

f¯
the average value of f on the interval
c3
the coefficient of x3 in f
c2
the coefficient of x2
c1
the coefficient of x
c0
the constant term
a
the start of the interval
b
the end of the interval (not equal to a)

When to use it

Use it for the average of something that varies continuously: average temperature over a day, average velocity from a velocity function, average power. It is the continuous version of adding values up and dividing by how many there are.

Worked example

Find the average value of f(x) = x² on [0, 3] (c₃ = 0, c₂ = 1, c₁ = 0, c₀ = 0, a = 0, b = 3).

  1. The average value is the integral divided by the width of the interval:

    f¯=1b−a∫abf(x)dx

  2. The function is

    f(x)=x2

  3. Integrate term by term with the power rule (raise each power by one and divide by the new power):

    F(x)=x33

  4. Evaluate at the upper limit:

    F(3)=333
    =273
    =9

  5. Evaluate at the lower limit:

    F(0)=033
    =03
    =0

  6. Subtract: the integral is F(b) − F(a).

    I=9−0
    =9

  7. Numerical check: Simpson's rule with 200 strips gives 9, which agrees.

  8. Divide by the width b − a:

    f¯=Ib−a
    =93−0
    =93
    =3

  9. A rectangle of height 3 over the same interval has the same area as the region under the curve.

Answer The average value is 3

Why it works

Split [a, b] into n equal pieces and average f at one point in each: 1n∑f(xi)=1b−a∑f(xi)Δx, because Δx=(b−a)/n. As n grows the sum becomes the integral.

Common mistakes

  • Averaging only the endpoint values: (0 + 9)/2 = 4.5 is wrong for x² on [0, 3].
  • Forgetting to divide by b−a.
  • Confusing it with the average rate of change f(b)−f(a)b−a, which is the slope of a secant.