Step by step
The question
Question: find the average value of f(x) = 16 − x² on [0, 4].
1 The formula
The average value: the area under the curve, divided by the width of the interval.
Here the width is 4 minus 0: 4.
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 16x − x³ over 3.
Put in 4, then take away the same with 0.
That is 128/3 minus 0: 128/3.
3 Divide by the width
Divide by the width, 4.
The average value is 32/3.
Careful!
128/3 is the area.
To get the average height, divide it by the width, 4.
The right way: average = 128/3 over 4 = 32/3.
4 Check
Let's check: a rectangle 32/3 high and 4 wide has area 128/3: the same as the integral.
Right!
Let's look back.
First: the formula.
Then: the integral.
Then: divide by the width.
Last: check.