402 Chapter 7 • The Riemann Integral
- Suppose f and g are continuous on [a, b] and l: f = l: g. Prove that 3 c E
[a, b] 3 f(c) = g(c). [See Exercise 5.1.26.] Find examples of discontinuous
f , g for which this conclusion is not true. - Prove the claim made in Example 7.5.3. (See Exercise 7.4.18.)
- Prove Corollary 7.5.6.
- Find a function f:[O, 1] -->JR such that f is integrable on [O, 1] and \Ix E
[0,1], f(x) > 0, but 1/f is not integrable on [0,1]. Does this contradict
Corollary 7.5.6 (b)? - Prove that if f:[a, b] -->JR is continuous and nonzero on [a, b], then 1/ f is
integrable on [a, b]. - Find functions f,g: [O, 1] -->JR such that f and fog are integrable on
[O, 1] but g is not.
12. Prove Corollary 7.5.7.13. Squeeze Principle: Suppose f , g : [a, b] --> JR are integrable on [a, b],
and \Ix E [a, b], f(x) :::; h(x) :::; g(x). Prove that if l: f = l: g, then his
integrable on [a, b] and l: h = l: f.- Suppose f:[a, b] --> JR is integrable on [a, b] and k E R Define g on [a+
k, b + k] by g(x) = f(x - k). Prove that g is integrable on [a+ k, b + k],
and l::: g = l: f. [That is, the integral is translation invariant.]
7 .6 The Fundamental Theorem of Calculus
So far in our development of the integral we have ignored antidifferentiation.
That is because the definition of the Riemann integral does not involve the
antiderivative in any way. We have not even mentioned the possibility that
there may be a connection between the integral and the derivative. But it is now
time to show the connection between these two great pillars of the calculus. The
fundamental theorem of calculus establishes that, in some sense, differentiation
and integration are inverse processes.
The "fundamental theorem of calculus" exists in two forms. The first form
is concerned with integrating derivatives. It is quite easily proved, and requires
only what we already know from the beginning sections of this chapter and
the definition of antiderivative. This form of the fundamental theorem is well
known to all students of calculus, since it is the basic tool used to calculate
integrals. Indeed, without this theorem, calculating integrals would be about
as difficult as calculating derivatives without any derivative formulas.