## M 408S,  INTEGRAL CALCULUS FOR SCIENCE

Text: Stewart, Calculus, Early Transcendentals, 8th Edition

Responsibile Parties: Jane Arledge, Kathy Davis, Ray Heitmann, December  2011.

Prerequisite and degree relevance: The prerequisite for M 408S is a grade of C- or better in either M408C, M308K, M408K, M308N, M408N, or an equivalent course from another institution; B in M408R  Only one of the following may be counted: M 403L, 408D, 408L, or 408S.

Calculus is offered in three equivalent sequences at UT: an accelerated two-semester sequence,  M 408C/D, and  two three-semester sequences, M 408K/L/M and M 408N/S/M. The latter is restricted to students in the College of Natural Sciences.

Completion (with grades of C- or better) of one of these calculus sequences is required for a mathematics major.  For some degrees, the two-semesters  M 408N/S satisfies the calculus requirement. These two courses are also a valid prerequisite for some upper-division mathematics courses, including M 325K, 427K, 340L, and 362K.

Course description:  M 408S is the second-semester calculus course of the three-course calculus sequence.  It is restricted to College of Natural Science Students.  It is an introduction to the theory and applications of integral calculus of functions of one variable.

The syllabus for M 408S includes most of the basic topics of integration on functions of a single real variable: the fundamental theorem of calculus, applications of integrations, techniques of integration, sequences, and infinite series.  The emphasis in this course is on problem solving, not on the presentation of theoretical considerations. While the course includes some discussion of theoretical notions, these are supporting rather than primary.

### Overview and Course Goals

The following pages comprise the syllabus for M 408S, and advice on teaching it.  Calculus is a service course, and the material in it was chosen after interdepartmental discussions.  Please do not make drastic changes (for example, skipping techniques of integration).  You will do your students a disservice and leave them ill equipped for subsequent courses.

This is not a course in the theory of calculus; the majority of the proofs in the text should not be covered in class.  At the other extreme, some of our brightest math majors found their first passion in calculus; one ought not to bore them.  Remember that 408N/S/M is the sequence designed for students who may not have taken calculus previously.  Students who have seen calculus and have done well might be better placed in the faster M 408C/408D sequence.

Resources for Students

Many students find the study skills from high school are not sufficient for UT.  The Sanger Learning Center (http://lifelearning.utexas.edu/) in Jester has a wide variety of material ( drills, video-taped lectures, computer programs, counseling, math anxiety workshops, algebra and trig review, calculus review) as well as tutoring options, all designed to help students through calculus.  On request they will come to your classroom and explain their services.

Timing and Optional Sections

A typical fall semester has 42 hours of lecture, 42 MWF and 28 TTh days, while a typical spring has 45 hours, 45 MWF and 30 TTh days (here, by one hour we mean 50 minutes -- thus in both cases there are three "hours" of lecture time per week).   The following syllabus contains suggestions as to timing, and includes approximately 36 hours of required material.  Even after including time for exams, etc., there will be some time for the optional topics, reviews, and/or additional depth in some areas.

### Syllabus

• Ch. 5 Integrals (4 hours)
• 5.3 The Fundamental Theorem of Calculus (review)
• 5.4 Indefinite Integrals and the Net Change Theorem
• 5.5 The Substitution Rule
• Ch. 6 Applications of Integration (2 hours)
• 6.1 Areas between Curves
• 6.2 Volumes
• 6.3 Volumes by Cylindrical Shells (optional)
• Ch. 7 Techniques of Integration (9 hours)
• 7.1 Integration by Parts
• 7.2 Trigonometric Integrals (light)
• 7.3 Trigonometric Substitution
• 7.4 Integration of Rational Functions by Partial Fractions
• 7.5 Strategy for Integration
• 7.7 Approximate Integration (optional)
• 7.8 Improper Integrals
• Ch. 9 Differential Equations (optional -- not in special UT version of book)
• 9.3 Separable Equations
• 9.4 Models for Population Growth
• Ch. 14 Partial Derivatives (1 hour)
• 14.3 Partial Derivatives
• Ch. 15 Multiple Integrals (4 hours)
• 15.1 Double Integrals over Rectangles
• 15.2 Double Integrals over General Regions
• Ch. 11 Infinite Sequences and Series (16 hours)
• 11.1 Sequences
• 11.2 Series
• 11.3 The Integral Test and Estimates of Sums
• 11.4 The Comparison Tests
• 11.5 Alternating Series
• 11.6 Absolute Convergence and the Ratio and Root Tests
• 11.7 Strategy for Testing Series
• 11.8 Power Series
• 11.9 Representations of Functions as Power Series
• 11.10 Taylor and Maclaurin Series
• 11.11 Applications of Taylor Polynomials (optional)