Topics In Analysis

3.5 Fundamental Theorem Of Calculus

In this section F

= x so things are specialized to the ordinary Riemann integral. With these properties, it is easy to prove the fundamental theorem of calculus² . Let f ∈ R

. Then by 3.4.10 f ∈ R

for each x ∈

. The first version of the fundamental theorem of calculus is a statement about the derivative of the function

\int x x \to f (t)dt. a

Theorem 3.5.1 Let f ∈ R

and let

\int x F (x) \equiv a f (t)dt.

Then if f is continuous at x ∈

' F (x) = f (x).

Proof: Let x ∈

be a point of continuity of f and let h be small enough that x + h ∈

. Then by using 3.4.13,

\int h-1(F (x+ h)- F (x)) = h-1 x+hf (t)dt. x

Also, using 3.4.11,

\int x+h f (x ) = h-1 f (x)dt. x

Therefore, by 3.4.15,

| | | -1 | || -1\int x+h || |h (F (x+ h)- F (x))- f (x)| = ||h x (f (t)- f (x))dt||

| | || \int x+h || \leq ||h-1 |f (t)- f (x)|dt||. x

Let ε > 0 and let δ > 0 be small enough that if

< δ, then

|f (t)- f (x)| < ε.

Therefore, if

< δ, the above inequality and 3.4.11 shows that

||h-1 (F (x+ h) - F (x))- f (x)|| \leq |h|-1 ε|h| = ε.

Since ε > 0 is arbitrary, this shows

hli\tom0 h-1(F (x+ h)- F (x)) = f (x)

and this proves the theorem.

Note this gives existence for the initial value problem,

F'(x) = f (x),F (a) = 0

whenever f is Riemann integrable and continuous.³

The next theorem is also called the fundamental theorem of calculus.

Theorem 3.5.2 Let f ∈ R

and suppose there exists an antiderivative for f,G, such that

G '(x ) = f (x)

for every point of

and G is continuous on

. Then

\int b f (x)dx = G (b)- G (a). (3.5.16) a

(3.5.16)

Proof: Let P =

be a partition satisfying

U (f,P) - L (f,P) < ε.

Then

G (b) − G (a)	= G (xn ) − G (x0)
	= ∑ _i=1ⁿG (xi) − G (xi−1) .

By the mean value theorem,

G (b) − G (a)	= ∑ _i=1ⁿG^′ (zi) (xi − xi−1)
	= ∑ _i=1ⁿf (zi) Δx_i

where z_i is some point in

. It follows, since the above sum lies between the upper and lower sums, that

G (b)- G (a) \in [L (f,P ),U (f,P )],

and also

\int b f (x) dx \in [L (f,P),U (f,P)]. a

Therefore,

| | || \int b || ||G (b)- G (a)- a f (x)dx|| < U (f,P) - L (f,P) < ε.

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

and F^′ = f on

. Then

\int b b f (x )dx = F (b)- F (a) \equiv F (x)|a. a

Definition 3.5.3 Let f be a bounded function defined on a closed interval

and let P ≡{x₀,

, x_n} be a partition of the interval. Suppose z_i ∈

is chosen. Then the sum

n\sum f (zi)(xi - xi-1) i=1

is known as a Riemann sum. Also,

||P|| \equiv max {|xi - xi-1| : i = 1,\cdot\cdot\cdot,n}.

Proposition 3.5.4 Suppose f ∈ R

. Then there exists a partition, P ≡

with the property that for any choice of z_k ∈

|\int n | || bf (x)dx- \sum f (z )(x - x )||< ε. ||a k=1 k k k- 1||

Proof: Choose P such that U

− L

< ε and then both ∫ _a^bf

dx and ∑ _k=1ⁿf

are contained in

and so the claimed inequality must hold. This proves the proposition.

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 said to be Riemann integrable if there exists a number, I with the property that for every ε > 0, there exists δ > 0 such that if

P \equiv {x0,x1,\cdot\cdot\cdot,xn}

is any partition having

< δ, and z_i ∈

| | || \sumn || ||I - f (zi)(xi - xi-1)|| < ε. i=1

The number ∫ _a^bf

dxis defined as I.

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

is Riemann integrable in the sense of Definition 3.5.5 if and only if it is integrable in the sense of Darboux. Furthermore the two integrals coincide.

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

is sufficiently small. Both versions of the integral are obsolete but entirely adequate for most applications and as a point of departure for a more up to date and satisfactory integral. The reason for using the Darboux approach to the integral is that all the existence theorems are easier to prove in this context.