Average value of f(x) = 3x^2 + 1 on [0, 3], and find c Step by Step

Step by step

The question

Question: find the average value of f(x) = 3x² + 1 on [0, 3], and the number c where f(c) is the average.

f(x) = 3x² + 1

1 The formula

The average value: the area under the curve, divided by the width of the interval.

average = 1b − a∫ from a to b of f(x) dx

Here the width is 3 minus 0: 3.

= 13 − 0∫ from 0 to 3 of (3x² + 1) dx

Think of it as the height of a rectangle with the same area as under the curve.

average height × width = area

2 The integral

The antiderivative: add 1 to each power, divide by the new power.

Now we have x³ + x.

∫ from 0 to 3 of (3x² + 1) dx = (x³ + x) from 0 to 3

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

= (3³ + 3) − (0³ + 0)

That is 30 minus 0: 30.

= 30 − 0 = 30

3 Divide by the width

Divide by the width, 3.

The average value is 10.

average = 303 = 10

Careful!

30 is the area.

To get the average height, divide it by the width, 3.

Careful: not average = 30 ✗

4 Where f(c) is the average

Now find c where f is exactly the average.

f(c) = 10

3c² + 1 = 10

Take away 1 on both sides.

3c² = 9

Divide by 3.

c² = 3

Take the square root: c is plus or minus √3, about 1.73.

c = ±√3 ≈ ±1.73

Only c between 0 and 3 counts: c = √3.

c = √3 ≈ 1.73

5 Check

Let's check: a rectangle 10 high and 3 wide has area 30: the same as the integral.

Right!

10 × 3 = 30 = 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.

Answer