Step by step
The question
Question: find the average value of f(x) = x² on [0, 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 0: 3.
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 over 3.
Put in 3, then take away the same with 0.
That is 9 minus 0: 9.
3 Divide by the width
Divide by the width, 3.
The average value is 3.
Careful!
9 is the area.
To get the average height, divide it by the width, 3.
The right way: average = 9/3 = 3.
4 Check
Let's check: a rectangle 3 high and 3 wide has area 9: the same as the integral.
Right!
In a picture:
the area under the curve,
and the rectangle at the average height.
The same area.
5 Pick the choice
Our answer is average = 3.
That is choice A.
6 Why not the others
B is 4.5.
Wrong: the mean of f(0) and f(3) is not the average value.
C is 9.
Wrong: did not divide by 3, the length of the interval.
That is the Careful mistake from before.
D is 27.
Wrong: multiplied by 3.
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.