7.6 The Fundamental Theorem of Calculus 403INTEGRATING DERIVATIVES
Definition 7.6.1 A function F is an antiderivative of a function f over a
set A if both f and Fare defined over A and Vx EA, F'(x) = f(x).Theorem 7.6.2 (Fundamental Theorem of Calculus, First Form) Sup-
pose f is integrable over [a, b]. If F is any antiderivative off over (a, b) that is
continuous over [a, b], thenI l: f = F(b) - F(a).1
Proof. Suppose f is integrable over [a, b], and F is an antiderivative of f
over (a, b) that is continuous over [a, b]. By the definition of antiderivative, F
is differentiable over (a, b), and Vx E (a, b), F'(x) = f(x).
Let P = {xo, x 1 , · · · , Xn} be any partition of [a, b]. Then the mean value
theorem can be applied to F over each subinterval [xi-l, xi], assuring us that
::i ::ixi * E ( Xi-1,Xi ) 3 F'( xi= *) F(xi) - F(xi-1) E. l l i: • 2
6.. qmvaenty,1ori=l, ,···n,
i
n
Thus, F(b) - F(a) = L [F(xi) - F(xi-1)]
i=l
[All terms cancel out except F(xn) and -F(xo).]
n
= °Lf(x:)6i
i=l
= R(f, P *), a Riemann sum off over P*.
By Lemma 7 .3.4, all Riemann sums lies between S_(f, P) and S(f, P), soS_(f, P) :::; F(b) - F(a) :::; S(f, P). (20)
But Pis an arbitrary partition of [a, b]. That means (20) is true for all par-
titions of [a, b]. By Theorem 7.2.15, there is only one number that lies between
every lower sum and every upper sum of an integrable function over [a, b], and
that is l: f. Therefore, l: f = F(b) -F(a). •
Remarks 7.6.3 Some caution should be exercised to avoid jumping to conclu-
sions not justified by the first form of the Fundamental Theorem of Calculus.
Theorem 7.6.2 does not assert that a function integrable over [a, b] has an an-
tiderivative there, nor does it assert that a function with an antiderivative over
[a, b] is integrable there. Indeed, there exist functions integrable over [a, b] that
do not have antiderivatives there (Exercises 3, 4, and 19) and functions that
have antiderivatives over [a, b] but are not integrable there (Exercise 6).