# Lower Sum, Upper Sum

Let . A partition of the interval is a collection of points

Let be a bounded function defined on and let be a partition of . Let

and

for . The lower sum of for is

and the upper sum of for is

We notice that: if , then

Therefore if and are partitions of with and , then the corresponding lower sums and upper sums satisfy the inequalities

In case partitions and of the same interval satisfy

it is possible to find partitions so that

with . The above observation shows now that the corresponding lower sums and upper sums satisfy the inequalities

Consider two arbitrary partitions and of . Then the partition

satisfies

The above observation now shows

Since it is always true that , we see that

for any two partitions , of . Hence

and

# Integral of a Bounded Function

A bounded function defined on is call integrable if there is exactly one number satisfying

for any partitions and of . This unique number is called the definite integral of on and is denoted

The above discussion shows that

# Characterization of an Integrable Function

Theorem. If is a bounded function defined on , then is integrable on if and only if for each there exists a partition of such that

# Every Continuous Function is Integrable

Theorem. If is continuous on , then is integrable on

Proof. Let . We show that there exists a partition of with . As was shown earlier the continuous function defined on the closed and bounded interval is uniformly continuous. Thus we can find such that

Choose any partition consisting of

with for all . It follows that

for . Therefore

# The First Fundamental Theorem of Calculus

Theorem. Let be integrable on . Define on by

If is continuous at in then is differentiable at , and

Proof. Let . We show that there is such that

whenever . Since is continuous at , there is a such that for

Case (i) . Since is a partition of the interval , it follows that

Therefore

Case (ii) . Since is a partition of the interval , it follows that

Therefore

# The Second Fundamental Theorem of Calculus

Theorem. If is integrable on and for some function , then

Proof. We need to show that

for every partition

of

By the Mean Value Theorem there is some point in such that

If

and

then

that is

Adding these equations for , we have

Therefore

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