MA126 Topics

Topic	Text Section	Subtopics	Comments
Antiderivatives	4.9	Definition of an Antiderivative   Antidifferentiation Formulas   Application: Rectilinear Motion	Introduces the concept of an antiderivative
Areas and Distances	5.1	Definition of Approximate Area Using Left and Right Approximating Rectangles   Application: The Distance Problem	Introduces the idea of defining the area under a curve using approximating rectangles and taking the limit as the number of rectangles becomes large.
The Definite Integral	5.2	Riemann Sums   Definition and Direct Evaluation of Definite Integrals   Using the Midpoint Rule to Approximate a Definite Integral   Algebraic Properties of the Definite Integral   Comparison Properties of the Definite Integral	Defines the definite integral as a limit of a sum, explores direct evaluation of this limit in a few special cases, and covers a number of important algebraic properties of definite integrals.
Evaluating Definite Integrals	5.3	The Evaluation Theorem   Definition of the Indefinite Integral   Application: The Net Change Theorem	Introduces the use of antiderivatives to evaluate definite integrals. Defines the indefinite integral and provides a short list of commonly used indefinite integrals.
The Fundamental Theorem of Calculus (FTC)	5.4	The FTC for a function defined in terms of integration limits   The FTC in terms of antiderivatives	The Fundamental Theorem of Calculus is the link between the two branches of calculus, differential calculus, which deals with derivatives, and integral calculus, which deals with integrals. Two statements of the theorem will be presented, one in terms of a function defined by limits of integration, and the second defined in terms of antiderivatives.
The Substitution Rule	5.5	Finding Antiderivatives using the Substitution Rule   The Substitution Rule for Definite Integrals   Some Symmetry Properties for Integrals	The Substitution Rule in a sense is the chain rule in reverse. It expands the class of functions for which we can find antiderivatives.
Integration by Parts	5.6	The Integration by Parts Formula   Integration by Parts for Definite Integrals	Integration by parts in a sense is the product rule for differentiation in reverse.
Additional Techniques of Integration	5.7 Appendix G	Trigonometric Integrals   Trigonometric Substitution   Partial Fractions	This section further expands our repertoire of integration techniques.
Integration Using Tables and Computer Algebra Systems	5.8	Integration Using Tables   Integration Using Computer Algebra Systems   Using the Midpoint Rule to Approximate a Definite Integral   Examples of Functions which Cannot Be Integrated in terms of Elementary Functions	Explores the use of tables and computers to find integrals, as well as the existence of elementary functions whose integrals are not elementary functions.
Numerical Integration	5.9	The Midpoint Rule   The Trapezoidal Rule   Simpson's Rule   Error Bounds	In this section we explore numerical techniques for approximate evaluation of definite integrals.
Improper Integrals	5.10	Infinite Intervals (Type I)   Discontinuous Integrands (Type II)   A Comparison Test for Convergence	Extends the definition of definite integrals to infinite intervals, and intervals containing discontinuities.
More About Areas	6.1	Areas Between Curves   Areas Enclosed by Parametric Curves	Some additional techniques for finding areas.
Volumes	6.2	Definition of the Volume of a Solid   Volumes of Revolution   The Method of Cylindrical Shells	Techniques for finding volumes of solid figures.
Arc Length	6.3	The Length of a Curve of the form y=f(x)   The Length of a Curve of the form x=f(y)   The Length of a Parametric Curve	Techniques for finding arc lengths.
The Average Value of a Function	6.4	Definition of the Average Value   The Mean Value Theorem for Integrals	Defines the average value of a function and presents techniques for finding it.
Applications to Physics and Engineering	6.5	Work   Hydrostatic Pressure and Force   Moments and Center of Mass	Some applications of integration in physics and engineering.
Applications to Economics and Biology	6.6	Consumer Surplus   Blood Flow   Cardiac Output	Some applications of integration in economics and biology.
Probability	6.7	Probability Density Functions   The Mean or Expected Value   The Median   The Normal or Gaussian Distribution	Applications of integration in the theory of probability.
Modeling with Differential Equations	7.1	Models of Population Growth   The Motion of a Spring   Differential Equations and Initial Value Problems	A brief introduction to differential equations.
Direction Fields and Euler's Method	7.2	Direction Fields   Euler's Method	Some graphical and approximation methods for studying differential equations without having an explicit solution.
Separable Equations	7.3	Definition of Separable Equations   Application: Mixing Problems	Techniques for a special class of differential equations.
Exponential Growth and Decay	7.4	Laws of Natural Growth and Decay   Application: Population Growth   Application: Radioactive Decay   Application: Newton's Law of Cooling   Application: Continuously Compounded Interest	Exponential growth and decay models and a few of their many applications.
The Logistic Equation	7.5	The Logistic Model   Approximate and Analytic Solutions   Other Population Growth Models	The Logistic Equation and some of its applications.
Predator-Prey Systems	7.6	The Lotka-Volterra Equations	A brief overview of a common biological population model and its dynamics.
Sequences	8.1	The Limit of a Sequence   Boundedness and Monotonicity	A brief introduction to sequences.
Series	8.2	Series and Partial Sums   Definition of Convergence for a Series   Geometric Series   Tests for Convergence and Divergence	A brief overview of infinite series.
The Integral and Comparison Tests	8.3	Testing Convergence with an Integral   The Comparison Test   The Limit Comparison Test   Estimating the Sum of a Series	Some techniques for determining whether a series converges or not. Some techniques for approximating the sum of a series.
Other Convergence Tests	8.4	The Alternating Series Test   Absolute Convergence   The Ratio Test	Some additional tests for convergence.
Power Series	8.5	Definition of a Power Series   Radius and Interval of Convergence	An introduction to power series.
Representing Functions as Power Series	8.6	Finding the Power Series Expansion of a Function   Differentiation and Integration of Power Series	Power series representation of functions.
The Taylor and Maclaurin Series	8.7	Definition of the Taylor Series   The remainder theorem and Taylor's inequality   Multiplication and Division of Power Series	An introduction to the Taylor and Maclaurin series.
The Binomial Series	8.8	Definition of the Binomial Series   The Fractional Case	The binomial series is useful for expanding integral and fractional powers of binomial and multinomial expressions.
Applications of the Taylor Polynomials	8.9	Approximating Functions by Polynomials   Applications to Physics	Some applications of Taylor Polynomials.