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# 16. Fundamental Theorem of Calculus

16.1   Definition (Nice functions.) I will say that a real valued function defined on an interval is a nice function on , if is continuous on and integrable on every subinterval of .

Remark: We know that piecewise monotonic continuous functions on are nice. It turns out that every continuous function on is nice, but we are not going to prove this. The theorems stated in this chapter for nice functions are usually stated for continuous functions. You can find a proof that every continuous function on an interval is integrable on (and hence that every continuous function on is nice on ) in [44, page 246] or in [1, page 153]. However both of these sources use a slightly different definition of continuity and of integral than we do, so you will need to do some work to translate the proofs in these references into proofs in our terms. You might try to prove the result yourself, but the proof is rather tricky. For all the applications we will make in this course, the functions examined will be continuous and piecewise monotonic so the theorems as we prove them are good enough.

16.2   Exercise. A Can you give an example of a continuous function on a closed interval that is not piecewise monotonic? You may describe your example rather loosely, and you do not need to prove that it is continuous.

16.3   Theorem (Fundamental theorem of calculus I.) Let be a nice function on . Suppose is an antiderivative for on . Then is an indefinite integral for on ; i.e.,
 (16.4)

Proof: By the definition of antiderivative, is continuous on and on . Let be arbitrary points in . I will suppose . (Note that if (16.4) holds when , then it holds when , since both sides of the equation change sign when and are interchanged. Also note that the theorem clearly holds for .)

Let be any partition of , and let be an integer with . If we can apply the mean value theorem to on to find a number such that

If , let . Then is a sample for such that

We have shown that for every partition of there is a sample for such that

Let be a sequence of partitions for such that , and for each let be a sample for such that

Then, since is integrable on ,

16.5   Example. The fundamental theorem will allow us to evaluate many integrals easily. For example, we know that . Hence, by the fundamental theorem,

This says that the two sets

and

have the same area - a rather remarkable result.

16.6   Theorem (Mean value theorem for integrals.) Let be a nice function on an interval , where . Then there is a number such that

Proof: Since is continuous on we can find numbers such that

By the inequality theorem for integrals

(here and denote constant functions) i.e.,

i.e.,

We can now apply the intermediate value property to on the interval whose endpoints are and to get a number between and such that

The number is in the interval , so we are done.

16.7   Corollary. Let be a nice function on a closed interval whose endpoints are and where . Then there is a number between and such that

16.8   Exercise. A Explain how corollary 16.7 follows from theorem 16.6. (There is nothing to show unless )

16.9   Lemma. Let be a function such that is integrable on every subinterval of . Let and let

Then is continuous on .

Proof: Let . I will show that is continuous at . Since is integrable on there is a number such that

By the corollary to the inequality theorem for integrals (8.17), it follows that

for all . Thus, for all ,

Now , so by the squeezing rule for limits of functions,
. It follows that .

16.10   Theorem (Fundamental theorem of calculus II.) Let be a nice function on , and let . Let

Then is an antiderivative for , i.e.
 (16.11)

In particular, every nice function on has an antiderivative on .

Proof: Let

and let be a point in . Let be any sequence in such that . Then

By the mean value theorem for integrals, there is a number between and such that

Now

and since , we have , by the squeezing rule for sequences. Since is continuous, we conclude that ; i.e.,

i.e.,

This proves that for . In addition is continuous on by lemma 16.9. Hence is an antiderivative for on .

Remark Leibnitz's statement of the fundamental principle of the calculus was the following:

Differences and sums are the inverses of one another, that is to say, the sum of the differences of a series is a term of the series, and the difference of the sums of a series is a term of the series; and I enunciate the former thus, , and the latter thus, [34, page 142].
To see the relation between Leibnitz's formulas and ours, in the equation , write to get , or . This corresponds to equation (16.11). Equation (16.4) can be written as

If we cancel the 's (in the next chapter we will show that this is actually justified!) we get . This is not quite the same as . However if you choose the origin of coordinates to be , then the two formulas coincide.

To emphasize the inverse-like relation between differentiation and integration, I will restate our formulas for both parts of the the fundamental theorem, ignoring all hypotheses:

By exploiting the ambiguous notation for indefinite integrals, we can get a form almost identical with Leibniz's:

16.12   Example. Let

We will calculate the derivatives of , and . By the fundamental theorem,

Now , so by the chain rule,

We have , so by the product rule,

16.13   Exercise. Calculate the derivatives of the following functions. Simplify your answers as much as you can.
a)
b)
c)
d)
e)
.
(We defined and in exercise 14.56.) Find simple formulas (not involving any integrals) for and for . A

16.14   Exercise. Use the fundamental theorem of calculus to find
a)
.
b)
.
c)
.

16.15   Exercise. Let and be the functions whose graphs are shown below:
Let for . Sketch the graphs of and . Include some discussion about why your answer is correct.

Next: 17. Antidifferentiation Techniques Up: Math 111 Calculus I Previous: 15.3 Convexity   Index
Ray Mayer 2007-09-07