Average value of f(x) = 3x^2 on [0, 2]

Step by step

The question

Question: find the average value of f(x) = 3x² 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 x³.

Put in 2, then take away the same with 0.

That is 8 minus 0: 8.

3 Divide by the width

Divide by the width, 2.

The average value is 4.

Careful!

8 is the area.

To get the average height, divide it by the width, 2.

4 Check

Let's check: a rectangle 4 high and 2 wide has area 8: the same as the integral.

Right!

Let's look back.

First: the formula.

Then: the integral.

Then: divide by the width.

Last: check.