Calculus: Single Variable Part 3 - Integration

About this course: Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach. In this third part--part three of five--we cover integrating differential equations, techniques of integration, the fundamental theorem of integral calculus, and difficult integrals.

Created by:  University of Pennsylvania

• Taught by:  Robert Ghrist, Professor

  Mathematics and Electrical & Systems Engineering

Commitment 6-8 hours/week Language English, Subtitles: Spanish How To Pass Pass all graded assignments to complete the course. Coursework

Each course is like an interactive textbook, featuring pre-recorded videos, quizzes and projects.

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University of Pennsylvania The University of Pennsylvania (commonly referred to as Penn) is a private university, located in Philadelphia, Pennsylvania, United States. A member of the Ivy League, Penn is the fourth-oldest institution of higher education in the United States, and considers itself to be the first university in the United States with both undergraduate and graduate studies.

Syllabus


WEEK 1


Integrating Differential Equations



Our first look at integrals will be motivated by differential equations. Describing how things evolve over time leads naturally to anti-differentiation, and we'll see a new application for derivatives in the form of stability criteria for equilibrium solutions.


5 videos, 2 readings, 4 practice quizzes expand

1. Reading: How Grading Works
2. Reading: Your Guide to Getting Started in this Course
3. Video: Indefinite Integrals
4. Practice Quiz: Challenge Homework: Indefinite Integrals
5. Video: A Simple O.D.E.
6. Practice Quiz: Challenge Homework: A Simple O.D.E.
7. Video: O.D.E.s
8. Practice Quiz: Challenge Homework: O.D.E.s
9. Video: O.D.E. Linearization
10. Video: BONUS!
11. Practice Quiz: Challenge Homework: O.D.E. Linearization

Graded: Core Homework: Indefinite Integrals
Graded: Core Homework: A Simple O.D.E.
Graded: Core Homework: O.D.E.s
Graded: Core Homework: O.D.E. Linearization

WEEK 2


Techniques of Integration
Since indefinite integrals are really anti-derivatives, it makes sense that the rules for integration are inverses of the rules for differentiation. Using this perspective, we will learn the most basic and important integration techniques.


6 videos, 4 practice quizzes expand

1. Video: Integration by Substitution
2. Practice Quiz: Challenge Homework: Integration by Substitution
3. Video: Integration by Parts
4. Practice Quiz: Challenge Homework: Integration by Parts
5. Video: Trig Substitution
6. Video: BONUS!
7. Practice Quiz: Challenge Homework: Trig Substitution
8. Video: Partial Fractions
9. Video: BONUS!
10. Practice Quiz: Challenge Homework: Partial Fractions

Graded: Core Homework: Integration by Substitution
Graded: Core Homework: Integration by Parts
Graded: Core Homework: Trig Substitution
Graded: Core Homework: Partial Fractions

WEEK 3


The Fundamental Theorem of Integral Calculus



Indefinite integrals are just half the story: the other half concerns definite integrals, thought of as limits of sums. The all-important *FTIC* [Fundamental Theorem of Integral Calculus] provides a bridge between the definite and indefinite worlds, and permits the power of integration techniques to bear on applications of definite integrals.


3 videos, 1 practice quiz expand

1. Video: Definite Integrals
2. Video: The F.T.I.C.
3. Video: BONUS!
4. Practice Quiz: Challenge Homework: The F.T.I.C.

Graded: Core Homework: Definite Integrals
Graded: Core Homework: The F.T.I.C.

WEEK 4


Dealing with Difficult Integrals



The simple story we have presented is, well, simple. In the real world, integrals are not always so well-behaved. This last module will survey what things can go wrong and how to overcome these complications. Once again, we find the language of big-O to be an ever-present help in time of need.


4 videos, 1 reading, 3 practice quizzes expand

1. Video: Improper Integrals
2. Video: BONUS!
3. Practice Quiz: Challenge Homework: Improper Integrals
4. Video: Trigonometric Integrals
5. Practice Quiz: Challenge Homework: Trigonometric Integrals
6. Video: Tables and Software
7. Practice Quiz: Challenge Homework: Tables and Software
8. Reading: About the Chpater 3 Exam

Graded: Core Homework: Improper Integrals
Graded: Core Homework: Trigonometric Integrals
Graded: Chapter 3: Integration - Exam