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

Step by step

The question

Question: find the average value of f(x) = x² + 2 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 x³ over 3 + 2x.

Put in 3, then take away the same with 0.

That is 15 minus 0: 15.

3 Divide by the width

Divide by the width, 3.

The average value is 5.

Careful!

15 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 2 on both sides.

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 5 high and 3 wide has area 15: the same as the integral.

Right!

Let's look back.

First: the formula.

Then: the integral.

Then: divide by the width.

Then: where f(c) is the average.

Last: check.