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This lesson marks that start of Unit 4: Integration -- Numerical Integration. This unit and the remaining ones in the course are all directed at the third driving question of the course: How can we calculate the volume enclosed by a surface? We'll get to that question in Unit 6. But first we need to review integration, discuss various techniques for integrating functions, and learn how to approximate definite integrals for the cases where we cannot find an antiderivative using those techniques. The first and last topics on that list are where we'll start in this lesson and focus in this unit.
Lesson Preview
In this lesson we'll switch gears to integration. In Module 1 we'll briefly review the Evaluation Theorem (sometimes also lumped into the Fundamental Theorem of Calculus) and then move on to discussing Riemann sums. Some Calculus 1 courses discuss this topic, but not all do, so no prior exposure to Riemann sums will be assumed. All the work we did with series will come in handy, as we'll see, since working with Riemann sums involve manipulations of sums, sigma notation, and in some cases taking limits of finite sums (like what we did early on in the series unit to determine if a series converges based on its partial sums). In Modules 2-3 we'll then learn how to use Riemann sums to approximate definite integrals.
Learn
Learn
The lesson notes below contain a learning plan with three stages -- Learn, Reflect, and Practice -- and guidance for what to do within each stage. Some tips for you as you work through this resource, and those that it points to:
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 Notes
Lesson 17.pdfVideo 1 (Example 17.2)
Video 1 (Example 17.2)
Interactive Illustration of Riemann Sums
Interactive Illustration of Riemann Sums
Reflect
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
Practice
Work through the practice problems suggested below to see how much of this lesson you've understood.
Lesson 17 PP.pdfPage updated
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