Step by step
The question
Question: find the average value of f(x) = 4 − x² on [0, 2].
1 The formula
The average value: the area under the curve, divided by the width of the interval.
Here the width is 2 minus 0: 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 4x − x³ over 3.
Put in 2, then take away the same with 0.
That is 16/3 minus 0: 16/3.
3 Divide by the width
Divide by the width, 2.
The average value is 8/3.
Careful!
16/3 is the area.
To get the average height, divide it by the width, 2.
4 Check
Let's check: a rectangle 8/3 high and 2 wide has area 16/3: the same as the integral.
Right!
In a picture:
the area under the curve,
and the rectangle at the average height.
The same area.
Let's look back.
First: the formula.
Then: the integral.
Then: divide by the width.
Last: check.