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# 8.2 Properties of the Integral

8.9   Definition (Operations on functions.) Let and be functions where are sets. Let . We define functions , , , and as follows:

Remark: These operations of addition, subtraction, multiplication and division for functions satisfy the associative, commutative and distributive laws that you expect them to. The proofs are straightforward and will be omitted.

8.10   Definition (Partition-sample sequence.) Let be an interval. By a partition-sample sequence for I will mean a pair of sequences where is a sequence of partitions of such that
, and for each in , is a sample for .

8.11   Theorem (Sum theorem for integrable functions.) Let be integrable functions on an interval . Then and are integrable on and

and

Proof: Suppose and are integrable on . Let be a partition-sample sequence for . If and , then

Since and are integrable, we have

By the sum theorem for sequences,

Hence is integrable and . The proof of the second statement is left as an exercise.

8.12   Notation ( ) If is integrable on an interval we will sometimes write instead of . The " in this expression is a dummy variable, but the " is a part of the notation and may not be replaced by another symbol. This notation will be used mainly in cases where no particular name is available for . Thus

means where is the function on defined by for all . The " here stands for difference, and is a ghost of the differences that appear in the approximations for the integral. The notation is due to Leibniz.

8.13   Example. Let

This function is integrable over every closed bounded subinterval of , since it is a sum of five functions that are known to be integrable. By several applications of the sum theorem for integrals we get

8.14   Exercise. A Calculate the following integrals.
a)
. Here .
b)
.
c)
.
d)
.
e)
.
f)
Here , and denotes a constant function.

8.15   Theorem (Inequality theorem for integrals.) Let and be integrable functions on the interval such that

Then

8.16   Exercise. Prove the inequality theorem for integrals. A

8.17   Corollary. Let be an integrable function on the interval . Suppose for all . Then

Proof: We have

Hence by the inequality theorem for integrals

Hence

It follows that

8.18   Theorem. Let be real numbers with , and let be a function from to . Suppose is integrable on and is integrable on . Then is integrable on and .

Proof: Since is integrable on and on , it follows that is bounded on and on , and hence is bounded on . Let be a partition-sample sequence for . For each in we define a partition of and a partition of , and a sample for , and a sample for as follows:

Then there is an index such that .

Let
 (8.19) (8.20)

We have
 (8.21)

where

Let be a bound for on . Then

Also,

Now

Since

it follows from the squeezing rule that and hence .

From equation (8.21) we have

 (8.22)

Since and , we see that is a partition-sample sequence on , and is a partition-sample sequence on . Since was given to be integrable on and on , we know that

and

Hence it follows from (8.22) that

i.e., is integrable on and

8.23   Corollary. Let be real numbers with , and let be a bounded function on . If the restriction of to each of the intervals is integrable, then is integrable on and

8.24   Definition (Spike function.) Let be an interval. A function is called a spike function, if there exist numbers and , with such that

8.25   Theorem (Spike functions are integrable.) Let be real numbers with and .Let

Then is integrable on and .

Proof: Case 1: Suppose Observe that is increasing on the interval and decreasing on the interval , so is integrable on each of these intervals. The set of points under the graph of is the union of a horizontal segment and a vertical segment, and thus is a zero-area set. Hence

By the previous theorem, is integrable on , and

.

Case 2: Suppose . Then by case 1 we see that is integrable with integral equal to zero, so by the sum theorem for integrals too.

8.26   Corollary. Let be real numbers with and . Let be an integrable function and let be defined by

Then is integrable on and .

8.27   Corollary. Let be an integrable function from an interval to . Let be a finite set of distinct points in , and let be a finite set of numbers. Let

Then is integrable on and . Thus we can alter an integrable function on any finite set of points without changing its integrability or its integral.

8.28   Exercise. Prove corollary 8.26, i.e., explain why it follows from theorem 8.25.

8.29   Definition (Piecewise monotonic function.) A function from an interval to is piecewise monotonic if there are points in with such that is monotonic on each of the intervals .

8.30   Example. The function whose graph is sketched below is piecewise monotonic.

8.31   Theorem. Every piecewise monotonic function is integrable.

Proof: This follows from corollary 8.23.

8.32   Exercise. A Let

Sketch the graph of . Carefully explain why is integrable, and find .

8.33   Example. Let Then

Hence is integrable on , and

8.34   Exercise. A Calculate the following integrals. Simplify your answers if you can.
a)
.
b)
. Here .
c)
. Here .
d)

Next: 8.3 A Non-integrable Function Up: 8. Integrable Functions Previous: 8.1 Definition of the   Index
Ray Mayer 2007-09-07