Step by step
The question
Question: find the average value of f(x) = x³ − x on [0, 2].
f(x) = x3 − x
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 2 minus 0: 2.
= 12 − 0∫ from 0 to 2 of (x3 − x) 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 x⁴ over 4 − x² over 2.
∫ from 0 to 2 of (x3 − x) dx = (x44 − x22) from 0 to 2
Put in 2, then take away the same with 0.
= (244 − 222) − (044 − 022)
That is 2 minus 0: 2.
= 2 − 0 = 2
3 Divide by the width
Divide by the width, 2.
The average value is 1.
average = 22 = 1
Careful!
2 is the area.
To get the average height, divide it by the width, 2.
Careful: not average = 2 ✗
4 Check
Let's check: a rectangle 1 high and 2 wide has area 2: the same as the integral.
Right!
1 × 2 = 2 = area
Let's look back.
First: the formula.
Then: the integral.
Then: divide by the width.
Last: check.
Answer
Careful: not average = 2 ✗