What is the average value of f(x) = 3x^2 + 1 on the interval [0, 2]? Step by Step

Step by step

The question

Question: find the average value of f(x) = 3x² + 1 on [0, 2].

f(x) = 3x² + 1

There are four choices.

First we solve it, the normal way.

Then we pick.

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 2 minus 0: 2.

= 12 − 0∫ from 0 to 2 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 2 of (3x² + 1) dx = (x³ + x) from 0 to 2

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

= (2³ + 2) − (0³ + 0)

That is 10 minus 0: 10.

= 10 − 0 = 10

3 Divide by the width

Divide by the width, 2.

The average value is 5.

average = 102 = 5

Careful!

10 is the area.

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

The right way: average = 10/2 = 5.

Careful: not average = 10 ✗

right: average = 102 = 5 ✓

4 Check

Let's check: a rectangle 5 high and 2 wide has area 10: the same as the integral.

Right!

5 × 2 = 10 = area

5 Pick the choice

Our answer is average = 5.

That is choice B.

B) 5

6 Why not the others

A is 10.

Wrong: did not divide by 2, the length of the interval.

That is the Careful mistake from before.

A) 10 ✗ did not divide by 2, the length of the interval

C is 7.

Wrong: the mean of f(0) = 1 and f(2) = 13.

C) 7 ✗ the mean of f(0) = 1 and f(2) = 13

D is 4.

Wrong: lost the + 1, which gives x.

D) 4 ✗ lost the + 1, which gives x

Let's look back.

First: the formula.

Then: the integral.

Then: divide by the width.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer