# Syllabus for Math 1540: “Integral Calculus"

Math 1530 (Differential Calculus) and Math 1540 (Integral Calculus) are 3-hour courses which, together, cover the material of the 5-hour Math 1550 (Differential and Integral Calculus), which is an introductory calculus course designed primarily for engineering majors and certain other technical majors.

The Math 1540 student is assumed to know about limits and derivatives of polynomial, exponential, logarithmic, and trigonometric functions, up through section 4.5 of Stewart's text (below). No prior exposure to integral calculus is assumed by the instructors of this class (though many students have had integral calculus before). The testing and evaluation of each class is entirely at the discretion of the instructor. No departmental policy exists on the use of sophisticated calculators in Math 1540; that decision is left to the instructor.

## Basic skills the students should acquire during the course

• Differentiation
1. Solve basic optimization problems
• Integration
1. Understand anti-derivatives and know the basic anti-derivative formulas
2. Have an understanding of the Riemann Integral as a limit of Riemann sums
3. Be able to use both parts of the Fundamental Theorem
4. Evaluate definite integrals using substitution
5. Find the area between two curves and the volumes of solids of revolution
6. Find arc lengths and areas of surfaces of revolution
7. Understand the Mean Value Theorem for Integrals

## Math 1540 text: "Calculus, Early Transcendentals," 8th Edition, by James Stewart

A specific section-by-section syllabus for the Stewart text and comments are shown below. A recommended set of homework problems is not provided. The textbook has a wide range of problems, from drill level through conceptual analysis. The instructor is urged to assign a broad range of problems from each section. Do not merely assign drill problems. The non-routine, challenging problems should form some part of each homework assignment.

## Syllabus

(Section 4.5 is the last section covered in Math 1530. Section 4.6 is skipped by Math 1530, 1540, and 1550. Math 1540 picks up where Math 1530 leaves off, viz., section 4.7.)

### Chapter 4: Applications of Differentiation

Section 4.7
Optimization Problems: The students will encounter some difficulty in this section. Many will have forgotten geometry, needed for many of the optimization problems. You should expect to spend more than one day on this section.
Section 4.8
Newton’s Method: This method of finding roots, while not robust, does offer the students reinforcements of the geometric interpretation of the derivative and the ideas of convergence. Demonstrations of cases where Newton’s Method fails to converge should be presented.
Section 4.9
Antiderivatives: A fairly standard presentation.

### Chapter 5: Integrals

Section 5.1
Areas and Distances: This section motivates the formal definition of the Riemann Integral in Section 5.2. This material is important, as it establishes the presence of sigma notation and left and right hand sums.  He also uses distance traveled over time to further motivate Riemann Sums.
Section 5.2
The Definite Integral: The instructor should carefully develop the definition of the Riemann Integral. This section also includes many of the basic properties of the definite integral. The instructor should try to ground much of this section in geometric terms. The author introduces the midpoint approximation in this section.
Section 5.3
The Fundamental Theorem of Calculus: The author introduces the idea of area functions. The student will not have seen these before. Be sure to explain the notion of positive and negative area at this time. The proof of the FTC is necessary for every calculus class. Spend enough time in class to do it right. Differentiation of the area function is harder for the students to remember than the integral evaluation result.
Section 5.4
Indefinite Integrals and the Net Change Theorem: The students should memorize the first small table of integrals in section 5.4. The author brings up displacement and total distance traveled as integrals.  The students will have trouble with total distance traveled because of the absolute value being applied to velocity.
Section 5.5
The Substitution Rule: Students have trouble with this topic. Since the idea of substitution is so important in Calculus II, the instructor should be very sure that his students receive adequate practice in this topic. The instructor should stress the need to change the limits of integration when evaluating a definite integral using substitution. You might seek out more problems from other sources to augment those in the book.

### Chapter 6: Applications of Integration

Section 6.1
Areas Between Curves: This is a fairly standard presentation. Be sure to demonstrate some problems involving integration with respect to y.
Section 6.2
Volumes: Many students have a great deal of difficulty with volume problems. The instructor should illustrate his examples with pictures (to the best of his ability). Be advised that even with marvelously drawn pictures the volumes by slicing problems represent some of the most challenging problems in Calculus I. The students do not “see” well in three dimensions. The instructor should be careful when assigning homework, as some of these slicing problems are difficult for the beginning student. In addition to slicing problems, this section introduces volumes of solids of revolution using washers and disks. Remind the students that Riemann Rectangles rotated around an axis perpendicular to the rectangle form disks or washers.
Section 6.3
Volumes by Cylindrical Shells: Similar to the above, many illustrations may make this topic easier for your students.  Remind the students that when a Riemann Rectangle is rotated around an axis parallel to the rectangle a shell is formed.
Section 6.4
Work: This fairly standard physic topic may actually take two days to cover. The students have more trouble with the fluid pumping problems than with the other problems in this section. Be careful with the units.
Section 6.5
Average Value of a Function: The average value is an easy topic. Students understand the average value well enough. The Mean Value Theorem for Integrals is presented but not proven. The instructor should consider showing the proof, since it is another nice application of the MVT for derivatives.

### Chapter 8: Further Applications of Integration

Section 8.1
Arc Length: Be advised that some of these integrals will be too difficult for the students since we are skipping Chapter 7. Of course, even if we covered Chapter 7 some arc length integrals are non-elementary.
Section 8.2
Area of a Surface of Revolution: Again, some of these integrals will be too difficult for the students since we are skipping Chapter 7. The revolutions about the y axis should also be discussed. Gabriel's horn, while it is an improper integral is an interesting example with which to tease the students.
Sections 8.3
Fluid Pressure and Center of Mass: These are standard topics from physics. The warning about troublesome integrals is again appropriate. Be certain of units. Remember, there is a difference between weight-density and mass-density.

Last updated May 11, 2020.