Step by step
The question
Question: find the average value of f(x) = 3x² − 2x on [1, 3].
f(x) = 3x2 − 2x
1 The formula
The average value: the area under the curve, divided by the width of the interval.
average = 1b − a∫ from a to b of f(x) dx
Here the width is 3 minus 1: 2.
= 13 − 1∫ from 1 to 3 of (3x2 − 2x) dx
Think of it as the height of a rectangle with the same area as under the curve.
average height × width = area
2 The integral
The antiderivative: add 1 to each power, divide by the new power.
Now we have x³ − x².
∫ from 1 to 3 of (3x2 − 2x) dx = (x3 − x2) from 1 to 3
Put in 3, then take away the same with 1.
= (33 − 32) − (13 − 12)
That is 18 minus 0: 18.
= 18 − 0 = 18
3 Divide by the width
Divide by the width, 2.
The average value is 9.
average = 182 = 9
Careful!
18 is the area.
To get the average height, divide it by the width, 2.
Careful: not average = 18 ✗
4 Check
Let's check: a rectangle 9 high and 2 wide has area 18: the same as the integral.
Right!
9 × 2 = 18 = area
Let's look back.
First: the formula.
Then: the integral.
Then: divide by the width.
Last: check.
Answer
Careful: not average = 18 ✗