Calculus Lab: Riemann Sums, Integrals, and Average Values

Goals

The goal of this project is for you to develop and explain the use of Riemann sums in application problems. The focus of your writing should be on clear descriptions and justifications of your methods. The lab contains two groups of questions for your consideration.

Acknowledgement

This lab is a slightly-edited version of a lab of the same name by Eugene Herman and Charles Jones.

Part 1: Average Temperature

When we say, "The average temperature today was 60 degrees," we clearly intend the single number 60 to represent the entire range of temperatures for the day. It is not so clear, however, how this number is to be computed. If we have a finite sample of temperatures, we can simply compute their average. For example, from the table of hourly temperatures (see Figure 1) in Des Moines, Iowa, on June 10, 1990, we can compute the average temperature by adding the 24 numbers and dividing by 24.

1. Compute this average. How would you compute the average temperature if the temperatures were recorded every half-hour instead of every hour? Explain why this average will usually not be the same as the average of the hourly temperatures.

Our intuition suggests that measuring the temperature more often should lead to a better estimate of the average temperature. So let's take this idea to the limit. Suppose the temperature at time t in hours (0 ≤ t ≤ 24) is T = f(t) in degrees Fahrenheit. (See the graph in Figure 1.) If the temperature is measured n times in 24 hours, say at times t₁, t₂, ..., t_n, the average of these temperatures is

Suppose the times t_i are equally spaced so that t_i - t_i-1 = Δt_i = 24/n. Then

(Equation 1)

which has the form of a Riemann sum multiplied by 1/24.

2. The graph in Figure 1 represents the temperature function f whose values at each hour are exactly the temperatures in the table. Use the graph to compute the Riemann sum of f(t) with n = 6 and f evaluated at right endpoints of subintervals. Then multiply by 1/24. (The answer should be close to. but not equal to, your answer in step 1.) If you had used n = 24 instead of n = 6, you would have gotten exactly the answer in step 1; explain why.

3. Give a careful argument to explain why

    

   should be the true average temperature over the 24-hour time period. Use Riemann sums and limits in your argument.

This group of questions addressed a special case of the more general concept of the average of a function. More formally, average of a function f(x) on the interval [a,b] is defined as

(Equation 2)

The next part of this lab applies this idea of functional average within a second context.

Part 2: Velocity and Distance

4. Approximate the distance traveled and the average velocity of an object whose velocity function is described by the table in Figure 2.

   

   5. Suppose you are given the velocity v(t) of an object at all times t, where a ≤ t ≤ b. Assume v(t) ≥ 0 for all t. Use Riemann sums and limits, as in Part 1. to derive a formula for the distance traveled. Explain and justify your derivation. Also describe how the concepts of distance and area are related. (This should follow from your derivation.)

   6. Paralleling the definition of the average value of a function in equation (2), the average velocity is

      

      Use the formula you derived in step 5 to explain why average velocity is also equal to Δs/Δt (change in position divided by change in time).

   Work to Turn In

   • Commentary for steps 1, 2, 3, 4, 5, and 6.
   • Computations for steps 1, 2, 4, and 5.

This document is available on the World Wide Web as

http://www.cs.grinnell.edu/~walker/courses/131.fa09/labs/lab-avg-value.shtml

created 9 June 2009 last revised 17June 2009
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.