Step by step
The question
Question: find the average value of f(x) = 3x² − 2x on [1, 3].
There are four choices.
First we solve it, the normal way.
Then we pick.
1 The formula
The average value: the area under the curve, divided by the width of the interval.
Here the width is 3 minus 1: 2.
Think of it as the height of a rectangle with the same area as under the curve.
2 The integral
The antiderivative: add 1 to each power, divide by the new power.
Now we have x³ − x².
Put in 3, then take away the same with 1.
That is 18 minus 0: 18.
3 Divide by the width
Divide by the width, 2.
The average value is 9.
Careful!
18 is the area.
To get the average height, divide it by the width, 2.
The right way: average = 18/2 = 9.
4 Check
Let's check: a rectangle 9 high and 2 wide has area 18: the same as the integral.
Right!
5 Pick the choice
Our answer is average = 9.
That is choice B.
6 Why not the others
A is 18.
Wrong: did not divide by 2, the length of the interval.
That is the Careful mistake from before.
C is 6.
Wrong: divided by 3: the length is 3 − 1 = 2.
D is 11.
Wrong: the mean of f(1) and f(3) is not the average value.
Let's look back.
First: the formula.
Then: the integral.
Then: divide by the width.
Then: check.
Then: pick the choice.
Last: why not the others.