Tuesday, January 3
−
Lecture 1:
Integration by substitution
(Refers to 6.1 in your
text)
After having practiced using the concepts of this lecture the student should be able
to: define the differential of a function, integrate
indefinite
integrals by making
appropriate change of variables (substitution), integrate
definite
integrals by
appropriate change of variables.
Summary of what we have learned about the notion of integration
.

Our study of integration began with attempts at finding the area of the region bounded
by the curve of
f
(
x
) and the
x
axis over an interval [
a
,
b
]. To do this we introduced
the notion of a Riemann sum. But computing areas in this way is inefficient.

The Fundamental theorem of calculus presented an alternate way to compute such
numbers. This important theorem is presented into two parts.

The second
part of the
Fundamental theorem of calculus
says that to find the area of
the region bounded by the curve of
f
(
x
) and the
x
axis over an interval [
a
,
b
] it suffices
to find an antiderivative of the function
f
(
x
),
evaluating it at the limits of integration
and subtracting the results. That is, if
f
(
x
) is continuous on [
a
,
b
] and
F
(
x
) is an anti
derivative of
f
(
x
) then

The first part of the
Fundamental theorem of calculus
says that, if
f
(
x
) is continuous
on [
a
,
b
] and
as
x
ranges over [
a
,
b
], then
This can also be expressed by

The first part of the FTC appears to be unrelated to our initial inquiry. In these notes
we used it to provide an easier proof of the second part of the FTC. But it is worth
remembering it since we occasionally invoke it in particular situations. It is a more
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 Winter '08
 ZHOU
 Calculus, Definite Integrals, Integrals, Integration By Substitution, 1.1.1.1

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