Chapter 7 Integration

The Fundamental Theorem of Calculus Section 7.3 The Fundamental Theorem of Calculus

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].

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

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

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

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

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 .

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 7.3.10 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