In this section F
|
Proof: Let x ∈
|
Also, using 3.4.11,
|
Therefore, by 3.4.15,
|
|
Let ε > 0 and let δ > 0 be small enough that if
|
Therefore, if
|
Since ε > 0 is arbitrary, this shows
|
and this proves the theorem.
Note this gives existence for the initial value problem,
|
whenever f is Riemann integrable and continuous.3
The next theorem is also called the fundamental theorem of calculus.
|
for every point of
| (3.5.16) |
Proof: Let P =
|
Then
G
− G | = G
− G |
= ∑
i=1nG
− G
. |
By the mean value theorem,
G
− G | = ∑
i=1nG′ |
= ∑
i=1nf Δ
xi |
where zi is some point in
|
and also
|
Therefore,
|
Since ε > 0 is arbitrary, 3.5.16 holds. This proves the theorem.
The following notation is often used in this context. Suppose F is an antiderivative of f as just described with F continuous on
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Definition 3.5.3 Let f be a bounded function defined on a closed interval
|
is known as a Riemann sum. Also,
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Proposition 3.5.4 Suppose f ∈ R
|
Proof: Choose P such that U
It is significant because it gives a way of approximating the integral.
The definition of Riemann integrability given in this chapter is also called Darboux integrability and the integral defined as the unique number which lies between all upper sums and all lower sums which is given in this chapter is called the Darboux integral . The definition of the Riemann integral in terms of Riemann sums is given next.
Definition 3.5.5 A bounded function, f defined on
|
is any partition having
|
The number ∫ abf
Thus, there are two definitions of the Riemann integral. It turns out they are equivalent which is the following theorem of of Darboux.
Theorem 3.5.6 A bounded function defined on
The proof of this theorem is left for the exercises in Problems 10 - 12. It isn’t essential that you understand this theorem so if it does not interest you, leave it out. Note that it implies that given a Riemann integrable function f in either sense, it can be approximated by Riemann sums whenever