Lecture 30
The definite integral
(Relevant section from Stewart, Sixth Edition: Section 5.2)
The material presented in this lecture closely followed the presentation of Section 5.2 of the text-
book, pp. 366-376.
Lecture 31
The Fundamental Theorem of Calculus
The material presented in this lecture, including the proofs of FTC I and II, closely followed the
presentation of Section 5.3 of the textbook, pp. 379-387.
This is perhaps one of the most important results of this course.
In both of the results stated
below (and proved in class), we assume that
f
(
x
) is a continuous function on
R
.
1. First define the function
g
(
x
) =
integraldisplay
x
a
f
(
t
)
dt.
(1)
Then
g
′
(
x
) =
f
(
x
)
.
(2)
This result is often known as the
First Fundamental Theorem of Calculus
, or simply “
FTC
I
.”
2. The above result establishes that the function
g
(
x
) is an
antiderivative
of
f
(
x
). The question
is, “Which one?” The answer is that
g
(
x
) is the antiderivative of
f
(
x
) for which
g
(0) = 0. From
the fact that
all
antiderivatives
F
(
x
),
f
(
x
) are related to
g
(
x
) as follows,
F
(
x
) =
g
(

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