Step by step
The question
Question: find the average value of f(x) = x² on [1, 4].
f(x) = x2
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 4 minus 1: 3.
= 14 − 1∫ from 1 to 4 of x2 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 over 3.
∫ from 1 to 4 of x2 dx = (x33) from 1 to 4
Put in 4, then take away the same with 1.
= (433) − (133)
That is 64/3 minus 1/3: 21.
= 643 − 13 = 21
3 Divide by the width
Divide by the width, 3.
The average value is 7.
average = 213 = 7
Careful!
21 is the area.
To get the average height, divide it by the width, 3.
Careful: not average = 21 ✗
4 Check
Let's check: a rectangle 7 high and 3 wide has area 21: the same as the integral.
Right!
7 × 3 = 21 = area
Let's look back.
First: the formula.
Then: the integral.
Then: divide by the width.
Last: check.
Answer
Careful: not average = 21 ✗