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# Averages Help

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By McGraw-Hill Professional
Updated on Aug 31, 2011

## Introduction to Averages

In ordinary conversation, when we average a collection p 1,..., pk of k numbers, we add them together and divide by the number of items:

The significance of the number σ is that if we wanted all the k numbers to be equal, but for the total to be the same, then that common value would have to be σ .

Now suppose that we want to average a continuous function f over an interval [ a, b ] of its domain. We can partition the interval,

P = { x 0 , x 1 ,..., x k },

with x 0 = a and x k = b as usual. We assume that this is a uniform partition , with x j − x j −1 = Δ x = ( b − a )/ k for all j . Then an “approximate average” of f would be given by

It is convenient to write this expression as

This last is a Riemann sum for the integral . Thus, letting the mesh of the partition go to zero, we declare

#### Example 1

In a tropical rain forest, the rainfall at time t is given by φ ( t ) = 0.1 − 0.1 t + 0.05 t 2 inches per hour, 0 ≤ t ≤ 10. What is the average rainfall for times 0 ≤ t ≤ 6?

#### Solution 1

We need only average the function φ:

#### Example 2

Let f ( x ) = x /2 − sin x on the interval [−2,5]. Compare the average value of this function on the interval with its minimum and maximum.

#### Solution 2

Observe that

Thus the critical points occur when cos x = 1/2, or at − π/3, π/3. We also must consider the endpoints −2, 5. The values at these points are

Plainly, the maximum value is f (5) = 5/2 − sin 5 ≈ 3.458924. The minimum value is f ( /3) ≈ −0.3424266.

The average value of our function is

You can see that the average value lies between the maximum and the minimum, as it should. This is an instance of a general phenomenon.

You Try It: On a certain tree line, the height of trees at position x is about 100 − 3x + sin 5x. What is the average height of trees from x = 2 to x = 200?

#### Example

What is the average value of the function g ( x ) = sin x over the interval [0, 2]?

#### Solution

We calculate that

We see that this answer is consistent with our intuition: the function g (x) = sin x takes positive values and negative values with equal weight over the interval [0, 2 ]. The average is intuitively equal to zero. And that is the actual computed value.

You Try It: Give an example of a function on the real line whose average over every interval of length 4 is 0.

Find practice problems and solutions for these concepts at: Applications of the Integral Practice Test.

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