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Chapter 4
Integration
Definition of an Antiderivative
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4-2
Theorem 4.1 Representation of Antiderivatives
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4-3
Basic Integration Rules
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4-4
Sigma Notation
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4-5
Theorem 4.2 Summation Formulas
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Figure 4.5
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Figure 4.6
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4-8
Figure 4.7
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4-9
Figure 4.8
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4-10
Figure 4.10
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4-11
Figure 4.11
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4-12
Figure 4.12
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4-13
Theorem 4.3 Limits of the Lower and Upper
Sums
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4-14
Definition of the Area of a Region in the Plane
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4-15
Definition of a Riemann Sum
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4-16
Definition of a Definite Integral
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4-17
Theorem 4.4 Continuity Implies Integrability
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4-18
Theorem 4.5 The Definite Integral as the Area
of a Region
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4-19
Definitions of Two Special Definite Integrals
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4-20
Theorem 4.6 Additive Interval Property
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4-21
Theorem 4.7 Properties of Definite Integrals
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4-22
Theorem 4.8 Preservation of Inequality
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Figure 4.27
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4-24
Theorem 4.9 The Fundamental Theorem of
Calculus
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4-25
Guidelines for Using the Fundamental
Theorem of Calculus
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4-26
Theorem 4.10 Mean Value Theorem for
Integrals and Figure 4.30
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4-27
Definition of the Average Value of a Function
on an Interval and Figure 4.32
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4-28
Definite Integral diagrams
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4-29
Figure 4.35
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4-30
Theorem 4.11 The Second Fundamental
Theorem of Calculus
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4-31
Theorem 4.12 Antidifferentiation of a
Composite Function
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4-32
Guidelines for Making a Change of Variables
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4-33
Theorem 4.13 The General Power Rule for
Integration
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4-34
Theorem 4.14 Change of Variables for Definite
Integrals
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4-35
Theorem 4.15 Integraion of Even and Odd
Functions and Figure 4.39
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4-36
Figure 4.41
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Theorem 4.16 The Trapezoidal Rule
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Theorem 4.17 Integral of p(x) =Ax2 + Bx + C
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4-39
Theorem 4.18 Simpson's Rule (n is even)
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4-40
Theorem 4.19 Errors in the Trapezoidal Rule
and Simpson's Rule
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