404 Chapter 7 • The Riemann Integral
To say this another way, Theorem 7.6.2 says that under certain conditions,J: f' = f(b) - f(a).
The conditions are that the integrand f' be a derivative and this derivative
must be integrable. As we have noted above, not every integrable function is a
derivative and not every derivative is integrable (Exercises 3, 4, 6, and 19).
DIFFERENTIATING INTEGRALSThe second form of the fundamental theorem is concerned with differenti-
ating integrals rather tha n integrating derivatives.
Up to now we have confined our attention to the integral of a function
over an interval with fixed endpoints. Indeed, our notation J: f suggests that
only the function f and the interval [a , b] are relevant, and that the integral
is a number, not a function. To see the connection between integration and
differentiation expressed by the second form of the fundamental theorem, we
must allow one of the endpoints of the interval of integration to be variable.
Specifically, we will be looking at the function F(x) = J: f.
So far, when we have written J: f we have implicitly assumed that a< b.
We will now allow a ~ b, but to do so requires a definition.
Definition 7.6.4 (a) \:/ a E IR, for any function f defined at a, we define
1: f = o.
(b) If f is integrable over [a, b], we define Jba f = - J: f.
Using these definitions, we are able to generalize Theorems 7.4.2 and 7.4.5,
as follows.
Theorem 7.6.5 If a, b, and c are any real numbers, then J: f = J: f + J: f,
regardless of the relative positions of a, b, and c, in the sense that if any two of
these integrals exist, then the third integral exists and this equation is satisfied.
Proof. Consider the three numbers a, b, and c.
Case 1: If two or more of the numbers a, b, and care equal, then J: f =J: f + J: f. (Exercise 1)
Case 2: Suppose a, b, and c are all different, and suppose f is integrable
on two of the intervals [a, b], [a, c], and [c, b]. There are six different possible
relative positions (permutations) of a, b, and c. We consider one of them, and
leave the other five for Exercise 2. Suppose c < b < a, and f is integrable on
two of the intervals [c, a], [c, b], and [b, a]. By Theorems 7.4.2 and 7.4.5, f is
integrable on the third, and
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