The average value of a function

Warm-up

Three quick questions from earlier lessons.

Discover

The violet curve is $y = x^2$ on $[0, 3]$ ; the blue line is $y = a$ . Move $a$ until the rectangle under the line on $[0, 3]$ has the same area as the region under the curve, $\displaystyle\int_0^3 x^2\,dx = 9$ .

The parabola y = x² on [0, 3] and a horizontal line y = a. At a = 3 the rectangle of width 3 under the line has area 9, the same as under the curve.

Which height $a$ works?

$a = 3$
$a = 4.5$

Key idea

The average value is the height of the rectangle with the same area.

The average value of

f

[a, b]

\dfrac{1}{b - a}\displaystyle\int_a^b f(x)\,dx

Worked example

Average value of $f(x) = 2x + 1$ on $[0, 4]$ .

\displaystyle\int_0^4 (2x + 1)\,dx = 20

F(x) = x² + x.

Average

= \tfrac{20}{4} = 5

Divide by the length 4.

Your turn, guided

Your turn: average of $f(x) = 3x^2$ on $[0, 2]$ .

Average

= 4

∫ = 8; divide by 2.

Practice ★

Find the average value of $f(x) = x^2$ on $[0, 3]$ .
Find the average value of $f(x) = x^2$ on $[0, 6]$ .
Find the average value of $f(x) = x^2$ on $[0, 2]$ .

Watch out!

Practice ★★

$v(t) = 3t + 6$ m/s. Find the average velocity on $[0, 8]$ , in m/s.
$v(t) = 3t + 1$ m/s. Find the average velocity on $[0, 7]$ , in m/s.

Find the average value of $f(x) = 6\sin x$ on $[0, \pi]$ . Give an exact value.

Exit ticket

Three questions to check you have got it.

Find the average value of $f(x) = x^2$ on $[0, 3]$ .
Find the average value of $f(x) = x^2$ on $[0, 6]$ .
Find the average value of $f(x) = x^2$ on $[0, 2]$ .

Summary card

Average value

$f_{\text{avg}} = \dfrac{1}{b - a}\displaystyle\int_a^b f(x)\,dx$ .

Watch out: divide by the length; do not just average the end values.

End of the lesson

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