Step by step
The question
Question: find the average value of f(x) = 9 − x² on [0, 3], and the number c where f(c) is the average.
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 9x − x³ over 3.
Put in 3, then take away the same with 0.
That is 18 minus 0: 18.
3 Divide by the width
Divide by the width, 3.
The average value is 6.
Careful!
18 is the area.
To get the average height, divide it by the width, 3.
4 Where f(c) is the average
Now find c where f is exactly the average.
Take away 9 on both sides.
Divide by −1.
Take the square root: c is plus or minus √3, about 1.73.
Only c between 0 and 3 counts: c = √3.
5 Check
Let's check: a rectangle 6 high and 3 wide has area 18: 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.
Then: where f(c) is the average.
Last: check.