Lesson 26: Riemann Integration
Preview
Today we’ll connect the process of finding local/global extrema to derivatives. We’ll also prove some of the foundational results related to derivatives that you learned about in calculus (e.g., the Mean Value Theorem).
In Module I we’ll build up to an Intermediate Value Theorem-like result called Rolle’s Theorem. Then, in Module II we’ll use Rolle’s Theorem to prove the Mean Value Theorem, an important result we’ll later use to prove the Fundamental Theorem of Calculus.
Review
Learn
Work through the lesson notes below. In class, I will fill in the "See Class Notes" boxes. Optional: Watch the video(s) below the embedded notes to get another instructor's take on the content covered in the notes.
Some tips for you as you work through these resources:
I recommend using Cornell Notes (or a modification of it; see this video starting at the 1:05 mark) to take notes on the lesson and the videos. This note-taking method balances detail with big-picture thinking to help you summarize and retain what you are learning. See this other video for additional note-taking techniques you might want to experiment with.
Lesson Notes
Lesson 26.pdfOptional Resources
Reflect
If you are currently enrolled in this course with me, submit the written reflections Google Form I have emailed you after working through the lesson notes and videos. Some tips:
Submit substantive, but concise, answers to each question; you will be doing the future you a big favor by taking time now to accurately and succinctly summarize what you have learned from the lesson.
Send yourself a copy of your reflections; they will come in handy later when you start preparing for quizzes and other assessments.
If you are not currently enrolled in this course with me, those written reflections ask three reflective questions designed to help you retain what you've learned and pinpoint any remaining areas of confusion. Those questions are:
Please summarize the main mathematical takeaways from the lesson notes.
What was the most interesting part of what you learned, and why?
What, if anything, do you still find confusing?
Practice
Work through the practice problems suggested below to see how much of this lesson you've understood. Book problems come from the book for the course, Mathematical Analysis, Second Edition, by Tom Apostol.
Lesson 26 PP.pdf