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# 7.1 Area Under a Monotonic Function

7.1   Theorem. Let be a monotonic function from the interval to . Let be a sequence of partitions of such that , and let

Then

and

(The notation here is the same as in theorem 5.40 and exercise 5.47.

Proof: We noted in theorem 5.40 and exercise 5.47 that

 (7.2)

Since

we conclude from the squeezing rule that
 (7.3)

We also have by (5.43) that

so that

By (7.3) and the squeezing rule

and hence

Also,

7.4   Definition (Riemann sum.) Let be a partition for an interval . A sample for is a finite sequence of numbers such that for . If is a function from to , and is a partition for and is a sample for , we define

and we call a Riemann sum for and . We will sometimes write instead of .

7.5   Example. If is an increasing function from to , and is a partition of , and , then

If , then

If then

is some number between and .

7.6   Theorem (Area theorem for monotonic functions.) Let be a
monotonic function from the interval to . Then for every sequence of partitions of such that , and for every sequence where is a sample for , we have

Proof: We will consider the case where is increasing. The case where is decreasing is similar.

For each partition and sample , we have for

Hence

i.e.,

By theorem 7.1 we have

and

so by the squeezing rule,

Next: 7.2 Calculation of Area Up: 7. Still More Area Previous: 7. Still More Area   Index
Ray Mayer 2007-09-07