**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.

, and for each in , is a sample for .

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.

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.

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

- a)
- . Here .
- b)
- .
- c)
- .
- d)
- .
- e)
- .
- f)
- Here , and denotes a constant function.

Proof: We have

Hence by the inequality theorem for integrals

Hence

It follows that

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 .

We have

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

and

Hence it follows from (8.22) that

i.e., 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.

Proof: This follows from corollary 8.23.

- a)
- .
- b)
- . Here .
- c)
- . Here .
- d)