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# 2.1 The Area Under a Power Function

Let be a positive number, let be a positive number, and let be the set of points in such that and . In this section we will begin an investigation of the area of .

Our discussion will not apply to negative values of , since we make frequent use of the fact that for all non-negative numbers and

Also is not defined when is negative.

The figures for the argument given below are for the case , but you should observe that the proof does not depend on the pictures.

Let be a positive integer, and for , let .

Then for , so the points divide the interval into equal subintervals. For , let

If , then for some index , and , so

Hence we have

and thus
 (2.1)

If , then and so . Hence, for all , and hence

so that
 (2.2)

Now

and

Since the boxes intersect only along their boundaries, we have
 (2.3)

and similarly

Thus it follows from equations (2.1) and (2.2) that

 (2.4)

The geometrical question of finding the area of has led us to the numerical problem of finding the sum

We will study this problem in the next section.

2.5   Definition (Circumscribed box.) Let be the smallest box containing . i.e.

Notice that . Thus equation (2.4) can be written as

 (2.6)

Observe that the outside terms in (2.6) do not depend on .

Now we will specialize to the case when . A direct calculation shows that

 (2.7)

There is a simple (?) formula for , but it is not particularly easy to guess this formula on the basis of these calculations. With the help of my computer, I checked that

Also

Thus by taking in equation (2.6) we see that

On the basis of the computer evidence it is very tempting to guess that

Next: 2.2 Some Summation Formulas Up: 2. Some Area Calculations Previous: 2. Some Area Calculations   Index
Ray Mayer 2007-09-07