Guided Practice for 4.2: Riemann sums

Overview

In this section we introduce a formalization of the idea developed in Section 4.1, where we estimated the distance traveled by a moving object by using a rectangle sum to estimate the area under the velocity curve. This concept is known as a Riemann sum, which is a geometric quantity that estimates the area between the graph of a function, the horizontal axis, and left and right bounds.

Learning objectives

Basic objectives: Each student is responsible for gaining proficiency with each of these tasks prior to engaging in class discussions, through the use of the learning resources (below) and through the working of exercises (also below).

• Calculate a rectangle sum approximation to the area under a curve (as in Preview Activity 4.2).
• Interpret sigma notation and use it to add up a sequence of numbers.
• Set up a left, right, or midpoint Riemann sum for a function on a given interval and then calculate that Riemann sum.

Advanced objectives: The following objectives are the subject of class discussion and further work; they should be mastered by each student during and following class discussions.

• Set up and calcualte Riemann sums for functions, by hand with technological assistance on the calculations.
• Use Riemann sums to answer questions about distance travelled, given information about velocity

Learning resources

To gain proficiency in the learning objectives, use the following resources. You may include other resources if you wish, in addition to or in replacement of the following.

Textbook: In Active Calculus, read Section 4.2. Make sure to read actively, working through examples and activities as you go.

Video: Watch the following videos at the MTH 201 YouTube playlist (http://bit.ly/GVSUCalculus).

• Quick review: Riemann sums (3:29)
• Sigma notation (7:15)
• Computing a left-hand Riemann sum (8:55)
• Another Riemann sum example (10:34)
• Calculating right and midpoint Riemann sums (9:33)

Activities

The following activity is to be done during and following your reading and viewing of the resources. Go to student.desmos.com and join using your name in the format Last, First. For example, I would enter my name as Ballif, Serge. Complete each part of the activity. Some of these problems will require you to work them out on paper before entering your answer. Practice producing high quality work so that your work is readable and meaningful. You will receive a mark of Pass if each item response shows a good-faith effort to be right and is submitted prior to the deadline. Remember to use the Piazza discussion board to ask about any questions you have.