Transcript title

Integral Calculus

Credits

4

Grading mode

Standard letter grades

Total contact hours

40

Lecture hours

40

Prerequisites

MTH 251Z (or higher) or minimum placement Math Level 24.

Course Description

This course explores Riemann sums, definite integrals, and indefinite integrals for real-valued functions of a single variable. These topics will be explored graphically, numerically, and symbolically in real-life applications. This course emphasizes abstraction, problem-solving, modeling, reasoning, communication, connections with other disciplines, and the appropriate use of technology.

Course learning outcomes

1. Approximate definite integrals using Riemann sums and apply this to the concept of accumulation and the definition of the definite integral.
2. Explain and use both parts of the Fundamental Theorem of Calculus.
3. Choose and apply integration techniques including substitution, integration by parts, basic partial fraction decomposition, and numerical techniques to integrate combinations of power, polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
4. Use the integral to model and solve problems in mathematics involving area, volume, net change, average value, and improper integration.
5. Apply integration techniques to solve a variety of problems, such as work, force, center of mass, or probability.

Content outline

1. Approximate definite integrals using Riemann sums and apply this to the concept of accumulation and the definition of the definite integral. 
   1. Students will be able to express finite sums using sigma notation.
   2. Students will be able to use Riemann sums to describe the process of approximating the net signed area between a curve and an axis.
   3. Students will be able to relate the definite integral with the concept of accumulation of area or other infinitesimal quantities, including the use of appropriate units.
2. Explain and use both parts of the Fundamental Theorem of Calculus.
   1. Students will be able to recognize and express the definite integral as a limit of a Riemann sum.
   2. Students will use and compare different methods for calculating definite integrals, such as linear properties of integrals, net-signed area, and graphical approaches.
   3. Students will explain and apply the concept of indefinite integrals and its connection to antidifferentiation.
   4. Students will explain the connection between derivatives and integrals and apply their understanding using the Fundamental Theorem of Calculus.
3. Choose and apply integration techniques including substitution, integration by parts, basic partial fraction decomposition, and numerical techniques to integrate combinations of power, polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
   1. Students will be able to integrate power, polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric functions using basic rules.
   2. Students will be able to use substitution and integration by parts to algebraically integrate appropriate combinations of functions.
   3. Students will be able to use partial fraction decomposition to evaluate integrals of rational functions whose denominators may be expressed as products of distinct linear factors.
   4. Students will be able to use numerical techniques, such as Midpoint, Trapezoidal, and Simpson’s rules, to approximate definite integrals.
4. Use the integral to model and solve problems in mathematics involving area, volume, net change, average value, and improper integration. 
   1. Students will be able to use definite integrals to find the area between two curves.
   2. Students will be able to calculate volumes of solids, such as solids of revolution or prisms, using integrals.
   3. Students will be able to apply the integral to find the average value of a function over an interval.
   4. Students will be able to apply the integral to find the net change of a function over an interval.
   5. Students will be able to recognize, describe, and calculate improper integrals.
5. Apply integration techniques to solve a variety of problems, such as work, force, center of mass, or probability.
   1. Students will apply integration to problems in the instructor’s choice of context, including but not limited to the possible options above. At least two distinct applications are recommended based on the population of students in the class.

Required materials

Students are required to have a license for web-based software which includes an e-text. Paper copy of the textbook is optional.

General education/Related instruction lists

• Mathematics