1. Chapter 7
Integration

2. The Fundamental Theorem of Calculus
Section 7.3
The Fundamental Theorem of Calculus

3. The Fundamental Theorem of Calculus I (Part 1)
Let f be integrable on [a, b]. For each x  [a, b], let
Then F is uniformly continuous on [a, b].
Proof:
Since f is integrable on [a, b], it is bounded there.
That is, there exists B > 0 such that | f (x) |  B for all x  [a, b].
To see that F is uniformly continuous on [a, b], suppose  > 0.
If x, y  [a, b] with x < y and | x – y | <  /B, then
Theorem 7.2.6
Corollary 7.2.8
Thus F is uniformly continuous on [a, b].

4. The Fundamental Theorem of Calculus I (Part 2)
Let
If f is continuous at c  [a, b], then F(x) is differentiable at c and F (c) = f (c).
Proof:
Given any  > 0 there exists a  > 0 such that | f (t) – f (c)| <  whenever
t  [a, b] and | t – c | < .
Now for all x  c we have
Thus for any x  [a, b] with 0 < | x – c | <  we have

5. Thus we have
Since  > 0 was arbitrary, we must have
In the Leibnitz notation:
Example 7.3.2 If
for x  0, then or

6. Corollary 7.3.3
Let f be continuous on [a, b] and let g be differentiable on [c, d], where
g([c, d])  [a, b]. Define
for all x  [c, d].
Then F is differentiable on [c, d] and
(The proof is a straightforward application of the chain rule.)
Example 7.3.4

7. The Fundamental Theorem of Calculus II
If f is differentiable on [a, b] and f ' is integrable on [a, b], then
Let P = {x0, x1, …, xn} be any partition of [a, b]. By applying the mean value theorem to each subinterval [xi – 1, xi], we obtain points ti  (xi – 1, xi) such that
Proof:
Thus we have
All but the first and last terms cancel.
Since mi( f ')  f '(ti)  Mi( f ') for all i, it follows that
L( f ', P)  f (b) – f (a)  U( f ', P).
This holds for each partition P, so we also have
L( f ' )  f (b) – f (a)  U( f ' ).
But f ' is assumed to be integrable on [a, b], so
Thus

8. Definition 7.3.8 Example 7.3.9(a)
Let f be defined on (a, b] and integrable on [c, b] for every c  (a, b]. If
exists, then the improper integral of f on (a, b], denoted by , is given by
If L = is a finite number, then the improper integral is said to converge to L.
If L =  (or – ), then the integral is said to diverge to  (or – ).
Example 7.3.9(a)
Let f (x) = x – 1/3 for x  (0, 1]. Then f is not integrable on [0, 1], since it is not bounded there. But for each c  (0, 1] we have
Taking the limit as c  0 +, we have
So the improper integral converges to .

9. Example 7.3.9(b) Definition 7.3.10
Let g (x) = 1/x for x  (0, 1]. Then for each c  (0, 1] we have
Since lim c  0 + (– ln c) = , the improper integral diverges to  and we write
We can also define an improper integral over an unbounded interval.
Definition
Let f be defined on [a, ) and integrable on [a, c] for every c > a. If
exists, then the improper integral of f on [a, ), denoted by , is given by