Chapter 4 Integration

Definition of an Antiderivative

Theorem 4.1 Representation of Antiderivatives

Basic Integration Rules

Sigma Notation

Theorem 4.2 Summation Formulas

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.10

Figure 4.11

Figure 4.12

Theorem 4.3 Limits of the Lower and Upper Sums

Definition of the Area of a Region in the Plane

Definition of a Riemann Sum

Definition of a Definite Integral

Theorem 4.4 Continuity Implies Integrability

Theorem 4.5 The Definite Integral as the Area of a Region

Definitions of Two Special Definite Integrals

Theorem 4.6 Additive Interval Property

Theorem 4.7 Properties of Definite Integrals

Theorem 4.8 Preservation of Inequality

Figure 4.27

Theorem 4.9 The Fundamental Theorem of Calculus

Guidelines for Using the Fundamental Theorem of Calculus

Theorem 4.10 Mean Value Theorem for Integrals and Figure 4.30

Definition of the Average Value of a Function on an Interval and Figure 4.32

Definite Integral diagrams

Figure 4.35

Theorem 4.11 The Second Fundamental Theorem of Calculus

Theorem 4.12 Antidifferentiation of a Composite Function

Guidelines for Making a Change of Variables

Theorem 4.13 The General Power Rule for Integration

Theorem 4.14 Change of Variables for Definite Integrals

Theorem 4.15 Integraion of Even and Odd Functions and Figure 4.39

Figure 4.41

Theorem 4.16 The Trapezoidal Rule

Theorem 4.17 Integral of p(x) =Ax2 + Bx + C

Theorem 4.18 Simpson's Rule (n is even)

Theorem 4.19 Errors in the Trapezoidal Rule and Simpson's Rule