Average value of f(x) = x^2 on [0, 6], and find c Step by Step

Step by step

The question

Question: find the average value of f(x) = x² on [0, 6], and the number c where f(c) is the average.

f(x) = x²

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 6 minus 0: 6.

= 16 − 0∫ from 0 to 6 of x² 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 over 3.

∫ from 0 to 6 of x² dx = (x³3) from 0 to 6

Put in 6, then take away the same with 0.

= (6³3) − (0³3)

That is 72 minus 0: 72.

= 72 − 0 = 72

3 Divide by the width

Divide by the width, 6.

The average value is 12.

average = 726 = 12

Careful!

72 is the area.

To get the average height, divide it by the width, 6.

Careful: not average = 72 ✗

4 Where f(c) is the average

Now find c where f is exactly the average.

f(c) = 12

c² = 12

Take the square root: c is plus or minus √12, about 3.46.

c = ±√12 ≈ ±3.46

Only c between 0 and 6 counts: c = √12.

c = √12 ≈ 3.46

5 Check

Let's check: a rectangle 12 high and 6 wide has area 72: the same as the integral.

Right!

12 × 6 = 72 = 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