Portland Community College | Portland, Oregon

Upon successful completion of the course, students should be able to:

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.

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1. Demonstrate an understanding of the concepts of integrals and their application to real world problems in:

   • at least two proctored, closed-book, no-notes exams worth at least 50% of the overall grade, one of which is a comprehensive final that is worth at least 25% of the overall grade, consisting primarily of free response questions (although a limited number of multiple choice and/or fill in the blank questions may be used where appropriate)

   • a closed book/closed note/no technology proctored exam over antiderivative formulae

   • laboratory reports (graded with an emphasis on proper notation and proper documentation) that account for at least 15% of the overall grade

   • And at least one of the following:

     • Take-home examinations

     • Quizzes

     • Graded homework problems

2. Consistently demonstrate proper notation, documentation, and use of language throughout all assessments and assignments. For proper documentation standards see Addendum.

3. Demonstrate an ability to work and communicate with colleagues, on the topics of antiderivatives and integrals, in at least two of the following:

   • A team project with a written report and/or in-class presentation

   • Participation in discussions

   • Group activities

• Students will learn to use the first and second derivatives of a function to find extreme function values and to solve applied maximum/minimum problems.

• Students will learn to evaluate indeterminate form limits using L'Hôpital's Rule.

• Students will learn the formal definition of the definite integral and several estimation techniques rooted in this definition.

• Students will learn to integrate function formulas and use the Fundamental Theorem of Calculus to evaluate definite integrals.

• Students will learn to model and solve several types of applications using definite integrals.

• Classroom activities will include lecture/discussion and group work.

• Students will communicate their results in oral and written form.

• Students will apply concepts to real world problems.

• The use of technology should be demonstrated and encouraged by the instructor where appropriate.

1. The Antiderivative and the Indefinite Integral The goal is to find the antiderivative(s) of a function expressed in graphical or symbolic form.

   1. Find the general antiderivative of functions that result in power, exponential, logarithmic, trigonometric, and inverse trigonometric functions. 

   2. Determine the particular antiderivative of a function given an initial value.

   3. Utilize the rules and strategies of integration.

      1. Utilize the sum, difference, and constant multiple rules of integration. 

      2. Integrate a function using substitution. 

      3. Integrate a function using integration by parts. 

      4. Integrate a function using partial fractions with denominators that may be expressed as distinct linear factors. 

      5. Find integrals using combinations of the above techniques. 

2. The Definite Integral The goal is to develop a practical understanding of the definite integral, to make connections between the derivative and the definite integral, and to understand multiple techniques for definite integral evaluation and estimation.

   1. Find left-hand and right-hand Riemann sums emphasizing the proper use of summation notation for functions presented in graphical, tabular, and/or symbolic form. 

   2. Evaluate a definite integral using a limit of Riemann sums (i.e. evaluate \(\displaystyle\int_a^b f(x)\ dx\) using the limit definition of the definite integral).

   3. Interpret the practical meaning of Riemann sums of rate functions. 

   4. Determine/estimate the net change in a function when the derivative of the function is presented in graphical, tabular, and/or symbolic form. 

   5. Explain the connection between definite integrals and net area between a function and the horizontal axis. 

   6. Explain and apply the connection between derivatives and integrals using the Fundamental Theorem of Calculus.

   7. Evaluate definite integrals using the Fundamental Theorem of Calculus and properties of the definite integral. 

   8. Approximate definite integrals numerically using techniques such as the Midpoint, Trapezoid, and Simpson’s Rules, and determine a bound for the error.

   9. Improper Integrals

      1. Determine divergence or convergence of improper integrals using limits and the Fundamental Theorem of Calculus.

      2. Determine the limit of a convergent improper integral. 

      3. Determine divergence or convergence of improper integrals by comparison.

3. Using the Integral The goal is to use the definite integral to solve application problems.

   1. Study applications to geometry.

      1. Find areas of planar regions using definite integrals.

      2. Calculate the volume of a solid using cross-sectional slices. 

      3. Calculate the volume of revolution generated by revolving a planar region around a line or axis using: 

         1. Disks

         2. Washers/Annulus

         3. Cylindrical Shells

      4. Represent the arc length of a curve as a definite integral and evaluate the integral symbolically, numerically, or using technology. 

   2. Apply the integral to find the average value of a function over an interval. 

   3. Use integrals to model and evaluate at least two applications in physics, engineering, or other sciences. 

      1. Work

      2. Center of mass 

      3. Hydrostatic force

      4. Cumulative distribution functions

      5. Probability

      6. Applications to biology and economics

4. Optional Topics

   1. Calculate the surface area of a surface over a closed region. 

   2. Integration with partial fractions that have repeated factors and irreducible quadratics.

All work in this course will be evaluated for your ability to meet the following writing objectives as well as for mathematical content.

1. Every solution must be written in such a way that the question that was asked is clear simply by reading the submitted solution.
2. Any table or graph that appears in the original problem must also appear somewhere in your solution.
3. All graphs that appear in your solution must contain axis names and scales. All graphs must be accompanied by a figure number and caption. When the graph is referenced in your written work, the reference must be by figure number. Additionally, graphs for applied problems must have units on each axis and the explicit meaning of each axis must be self-apparent either by the axis names or by the figure caption.
4. All tables that appear in your solution must have well defined column headings as well as an assigned table number accompanied by a brief caption (description). When the table is referenced in your written work, the reference must be by table number.
5. A brief introduction to the problem is almost always appropriate.
6. In applied problems, all variables and constants must be defined.
7. If you used the graph or table feature of your calculator in the problem solving process, you must include the graph or table in your written solution.
8. If you used some other non-trivial feature of your calculator (e.g., SOLVER), you must state this in your solution.
9. All (relevant) information given in the problem must be stated somewhere in your solution.
10. A sentence that orients the reader to the purpose of the mathematics should usually precede symbol pushing.
11. Your conclusion shall not be encased in a box, but rather stated at the end of your solution in complete sentence form.
12. Remember to line up your equal signs.
13. If work is word-processed, all mathematical symbols must be generated with a math equation editor.

This class is a foundational course for many STEM majors. Some topics are of particular importance for students continuing into MTH 253Z including: using L'Hôpital's Rule to evaluate limits, improper integrals, and error estimates for definite integrals. Students may be taking this course concurrently with calculus-based physics courses. It can be beneficial for these students if the integral symbol is introduced early on to represent antiderivatives.  Partial fractions are a particularly important technique for engineering students (which will be revisited in MTH 253Z and MTH 256). Students should be able to do simple partial fraction expansions by hand, but may use technology for more complicated problems. Since this course is also a prerequisite for MTH 261, logic and correct application of theorems should be emphasized.

Lab time shall be used by students to work on group activities—the activities to be used during lab are on the mathematics department home page at https://www.pcc.edu/programs/math/course-downloads.html

Emphasis should be placed on using technology such as Desmos and GeoGebra appropriately; such as when computing approximations, graphing curves, or visualizing or checking answers.  Technology should not be used as a substitute for meeting the outcomes and skills for the course that are expected to be done by hand.