Tuesday, November 15
−
Lecture 27 :
Integration by substitution
(Refers to Section
6.1 your text)
Students who have mastered the content of this lecture know
:
About differentials, integration by change
of variable (equivalently, by substitution).
Students who have practiced the techniques presented in this lecture will be able to
:
Compute integrals
by substitution correctly.
Summary of what 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
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The first part of the FTC appears to be unrelated to our initial inquiry. In these notes
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 Spring '11
 RobertAndre
 Calculus, Integration By Substitution

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