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# 7.2 Calculation of Area under Power Functions

7.7   Lemma. Let be a rational number such that . Let be a real number with . Then

(For the purposes of this lemma, we will assume that the limit exists. In theorem 7.10 we will prove that the limit exists.)

Proof: Let be a generic element of . To simplify the notation, I will write

Let

and let

Then for

so

It follows by the th root rule (theorem 6.48) that . Hence it follows from theorem 7.6 that
 (7.8)

Now
 (7.9)

Here we have used the formula for a finite geometric series. Thus, from (7.8)

Now we want to calculate the limit appearing in the previous lemma. In order to do this it will be convenient to prove a few general limit theorems.

7.10   Theorem. Let be a sequence of positive numbers such that and for all . Let be any rational number. Then

Proof: Suppose for all , and .

Case 1: Suppose . Then the conclusion clearly follows.

Case 2: Suppose . Then by the formula for a geometric series

By the sum theorem and many applications of the product theorem we conclude that

Case 3: Suppose . Let . Then , so by Case 2 we get

Case 4: Suppose where and . Let . Then

Now if we could show that , it would follow from this formula that

The next lemma shows that and completes the proof of theorem 7.10.

7.11   Lemma. Let be a sequence of positive numbers such that , and for all . Then for each in , .

Proof: Let be a sequence of positive numbers such that . Let for each in . We want to show that . By the formula for a finite geometric series

so

Now

Since , we have , so by the squeezing rule , and hence

7.12   Lemma (Calculation of .) Let be a real number with , and let . Then

Proof: By lemma 7.7,

By theorem 7.10,

and putting these results together, we get

7.13   Lemma. Let , and let , with . Then

Proof: If

is a partition of , let

be the partition of obtained by multiplying the points of by . Then
 (7.14)

If

is a sample for , let

be the corresponding sample for . Then

Let be a sequence of partitions of such that , and for each let be a sample for . It follows from (7.14) that . By the area theorem for monotonic functions (theorem 7.6), we have

Thus

7.15   Theorem (Calculation of .) Let with , and let . Then

Proof: The result for the case was proved in theorem 5.76. The case is done in the following exercise.

7.16   Exercise. A Use the two previous lemmas to prove theorem 7.15 for the case .

Remark: In the proof of lemma 7.7, we did not use the assumption until line (7.9). For equation (7.9) becomes

Since in this case ,we conclude that
 (7.17)

This formula give us method of calculating logarithms by taking square roots. We know will be near to when is large, and can be calculated by taking successive square roots. On my calculator, I pressed the following sequence of keys

and got the result . My calculator also says that
. It appears that if I know how to calculate square roots, then I can calculate logarithms fairly easily.

7.18   Exercise. A Let be a non-negative rational number, and let . Show that

Where in your proof do you use the fact that ?

Next: 8. Integrable Functions Up: 7. Still More Area Previous: 7.1 Area Under a   Index
Ray Mayer 2007-09-07