# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

## MAT 21B: Calculus: Integral Calculus

Approved: 2007-04-01 (revised 2021-12-08, J.Challenor and UPC)
ATTENTION:
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Thomas' Calculus Early Transcendentals, 15th Edition by George B. Thomas, Maurice Weir, and Joel Hass; Addison Wesley Publishers.
Prerequisites:
(MAT 021A C- or better or MAT 021AH C- or better) or MAT 017A B or better.
Suggested Schedule:
 Lecture(s) Sections Comments/Topics 1 4.8 Antiderivatives 1 5.1 Area and estimating with finite sums 1 5.2 Sigma notation and limits of finite sums 1 5.3 The definite integral 1.5 5.4 The Fundamental Theorem of Calculus 1 5.5 Indefinite integrals and the substitution method 1 5.6 Definite Integral Substitutions and the area between curves 1 6.1 Volumes using cross sections 1 6.2 Volumes using cylindrical shells 1 6.3 Arc length 1 6.4 Areas of surfaces of revolution 1 7.1 The Lorgarithm Defined as an integral 0.5 8.1 Using basic integration formulas 1 8.2 Integration by Parts 1 8.3 Trigonometric Integrals 1 8.4 Trigonometric Substitutions 1 8.5 Integration of rational functions by partial fractions 1 8.7 Numerical integration 2 8.8 Improper integrals 0.5 11.1 Parametrization of plane curves 1 11.2 Calculus with plane curves 0.5 11.3 Polar coordinates 1 11.4 Graphing Polar Coordinate Equations
Total number of lectures = 23. This leaves time for exams and time adjustments. Sections 8.8 and 11.1-4 are considered high priority and must be covered with appropriate attention. If time permits, instructors may want to integrate the following sections into their course:
• 7.2 Exponential Change and Separable Differential Equations
• 8.6 Integral Tables and Computer Algebra Systems
• 9.2 First-Order Linear Differential Equations
• 11.5 Areas and Lengths in Polar Coordinates

Instructors may want to consider the following adjustments to the schedule as stated above:
• Instructors may want to cover antiderivatives (section 4.8) after introducing Riemann sums and right before the Fundamental Theorem of Calculus (section 5.4). They may also want to wait to introduce indefinite integral notation until section 5.5.
• Instructors may find that they can discuss a wider variety of applications of integrals if the integration techniques in chapter 8 are covered before the applications in chapter 6.
• Instructors may consider adding an additional half-lecture on numerical integration error analysis in section 8.7.
• Covering section 11.5 (areas and lengths in polar coordinates) immediately after section 11.4 is a particularly appropriate way to conclude the class as it revisits the idea of a Riemann sum in the context of polar functions.
Learning Goals:
A goal of this course is to help students develop effective strategies for solving both mathematical and real-world problems. Although students often do not like “word problems” probing applications of their mathematical skills, it is very important that instructors emphasize these types of problems so that students become experts at them. In particular, students should be taught how to create mathematical models and develop effective strategies for solving problems in applied settings and non-routine situations. Students will learn the fundamentals of integral calculus. The course begins with the integral of a continuous function defined as the limit of its Riemann sum and then the Fundamental Theorems of Calculus presents integrals as antiderivatives. Integrals are applied to many problems in physics, including area, volume, arc length, and surface area. Various techniques of integration are studied and include substitution, integration by parts, integration by partial fractions, and trigonometric substitution. After discussing numerical integration and improper integrals, the course concludes with the calculus of parametric equations and polar coordinates. This course is a prerequisite for multivariable and vector calculus. Mastery of this course would be manifested in improved reading, writing, thinking, and problem-solving skills. Students should have an increased ability to understand, visualize, categorize, model, and solve complicated calculus problems in both two- and three-dimensional space.
Assessment:
Mastery of this course is usually assessed by periodic quizzes, homework problems, TA led discussion sections, hour exams, and a comprehensive final exam.