Calculus Group Activity: Fundamental Theorem of Calculus

Goals

This lab has three main goals:

1. Assist in clarifying the logical reasoning that leads to the Fundamental Theorem of Calculus.
2. Provide practice in reading mathematical arguments and synthesizing the progression of ideas behind mathematical results.
3. Aid in the understanding of both parts of the Fundamental Theorem of Calculus.

Background Reading

Sections 5.1, 5.2, and 5.3 of Stewart's Calculus (Sixth Edition) present a progression of ideas that leads to the Fundamental Theorem of Calculus.

Section 5.1

Section 5.1 describes sums involving left endpoints (L_n) and right endpoints (R_n). The first part of this Group Activity examines elements of these sums in additional detail.

Example 2 (pages 291-293) discusses approximating the area under the function f(x) = x² between x = 0 and x = 1 by dividing the interval [0, 1] into n strips and using right endpoints to obtain an expression for R_n.

1. Write the corresponding expression for L_n. That is, write an expression that gives the approximate area under the function f(x) = x² between x = 0 and x = 1, obtained by dividing the interval [0, 1] into n strips and using left endpoints to obtain an expression for L_n.

The textbook (page 291) uses the formula

to conclude that R_n = (n+1)(2n+1)/(6n²)

2. Compute a parallel expression for L_n.

The textbook (still p. 291) observes that the actual area under y = x² can be obtained by taking more and more rectangles, so the actual area is obtained by taking the limit of R_n as n goes to infinity. The text then uses the expression for R_n to compute this limit as 1/3 for the area under y = x² for x between 0 and 1.

3. Using your expression for L_n from Step 2, compute the limit of L_n as n goes to infinity. (One might expect that this limit also should be 1/3, but be sure to show enough algebra to explain why this result might be true.)

General Context

For the rest of this lab, assume that f(x) is a function defined on the interval [a, b], and assume that both L_n and R_n refer to sums to approximate the area under y = f(x) between x = a and x = b.

Non-decreasing Functions

For steps 4-8, assume that f(x) is a non-decreasing function. Roughly, this means that f(x) gets bigger as x gets bigger. More precisely, assume that f(u) ≤ f(v) whenever u ≤ v.

4. Assuming that f is a non-decreasing function, explain why
   1. the area estimate for L_n will always be less than or equal to the actual area, and
   2. the area estimate for R_n will always be greater than or equal to the actual area.
   That is, the actual area for non-decreasing functions will always be between L_n and R_n.

Since the actual area is between L_n and R_n, we can ask how the estimate for L_n differs from the estimate for R_n.

5. Compute the difference R_n - L_n explicitly as an expression involving f(x), a, b, and n.
6. Show that the difference R_n - L_n goes to 0 as n goes to infinity.
   (Hint: use step 5.)
7. Using step 6, explain
   1. why the limit of sums using L_n is the same as the limit of sums using R_n, and
   2. why the limit of sums using either L_n or R_n must give the actual area.

The textbook claims (at the top of page 294) that the area of a region under a continuous function can be computed with either left or right endpoints.

8. Does the conclusion in steps 1-7 above follow for any non-decreasing function, or is it necessary to conclude that the limit of the left and right sums will be the same for non-decreasing functions only if the functions are also continuous? Explain.

Section 5.2

Section 5.1 presents a lengthy discussion of left and right sums as a means to approximate the area under a curve.

9. Section 5.2 begins with a Definition of a Definite Integral. To what extent is anything in this Definition new or different from Section 5.1? Is this just a restatement of material from Section 5.1, or is something added here? Explain.

Section 5.3: Ideas Behind Results

Section 5.3 discusses Part I and Part II of the Fundamental Theorem of Calculus.

10. Give an example that illustrates Part I of the Fundamental Theorem of Calculus, and give an example that illustrates Part II.

Review the logical progression of ideas behind Part I and Part II of the Fundamental Theorem of Calculus.

11. For each Part of the Fundamental Theorem of Calculus, indicate the main idea behind why the statement is true. That is, summarize the main idea behind each argument in 3-5 sentences.

Work to Turn In

You may work on this extra-credit activity either individually or in groups of 2 or 3.

• If you work in a group, only one write-up should be turned in for the group, and each person's name in the group should appear at the top of the paper.
• Write-ups for each step should be clearly written and placed in a logical order (e.g., answers to Step 1, then Step 2, and then Step 3).
• You may talk to others about the ideas in this Group Activity, but the group should write the results up in its own words. Also, as with all other work for this course, a report must cite any ideas or contributions coming from consultations with others outside the group.

This document is available on the World Wide Web as

http://www.walker.cs.grinnell.edu/courses/131.fa12/groups/fund-theorem.shtml

created 5 November 2012 last revised 6 November 2012
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.