Step by step
The question
Question: find the average value of f(x) = 6x − x² on [0, 6].
1 The formula
The average value: the area under the curve, divided by the width of the interval.
Here the width is 6 minus 0: 6.
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 3 x² − x³ over 3.
Put in 6, then take away the same with 0.
That is 36 minus 0: 36.
3 Divide by the width
Divide by the width, 6.
The average value is 6.
Careful!
36 is the area.
To get the average height, divide it by the width, 6.
4 Check
Let's check: a rectangle 6 high and 6 wide has area 36: 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.