Step by step
The question
Question: find the average value of f(x) = x² + 1 on [0, 3].
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 + x.
Put in 3, then take away the same with 0.
That is 12 minus 0: 12.
3 Divide by the width
Divide by the width, 3.
The average value is 4.
Careful!
12 is the area.
To get the average height, divide it by the width, 3.
4 Check
Let's check: a rectangle 4 high and 3 wide has area 12: 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.