M CC 160 Calculus for Physical Scientists I

Calculator Lab: Calculating Riemann SumsPage1 of7

Name: ______MATH 160 Calculus for Physical Scientists I

Fall, 2008

Section: ______Calculator Exploration

Date due: ______

Calculator: ______

Calculating Riemann Sums

Overview The Riemann integral is defined as a limit of Riemann sums (see Sections 5.2 and 5.3 of the textbook). The limiting process in the definition of the Riemann integral is complicated. It is impractical, if not impossible, to evaluate most Riemann integrals with pencil and paper from the definition. However, it is possible, and even practical, to approximate most Riemann integrals very accurately from the definition by using a calculator.

In this lab you will learn to use your calculator to calculate Riemann sums efficiently. In the next lab you will use these calculator techniques to investigate how the lengths of the subintervals and the selection of the evaluation points used in the Riemann sum affect the accuracy with which a Riemann sum approximates an integral.

The investigations in this lab require a calculator that can produce traceable graphs. While many makes and models of calculators have these capabilities, the author used Texas Instrument calculators as he wrote this lab. The lab does not includecomplete instructions for using a calculator. Use the manual for your calculator to learn how to perform the tasks in this lab efficiently and accurately. Manuals for Texas Instrument calculators can be read from the Texas Instrument web site. Go to You can find instructions for many different calculators at . You might also search for manuals for other calculators on the manufacturers’ web sites.

The calculator skills you develop doing this lab will serve you well throughout this and other courses. If you encounter difficulties, take your calculator and manual to your instructor and discuss the problem with him/her. Classmates may be able to help out, too

The following factors will be considered in scoring your lab report:

•Completeness. Each investigation must be completed entirely, recorded fully, and explained or interpreted clearly.

•Mathematical and computational accuracy.

•Clarity and readability. Explanations must be written in complete sentences with correct spelling, capitalization, and punctuation and with reasonable margins and spacing. Handwriting must be legible. Tables and graphs must be presented in a clear, readable format.

Space for writing your report is provided within the lab. However, if you wish to word process your lab report, your instructor will e-mail you a copy of this lab as an attached MS Word document. Submit your final lab report as a printed document.

Space for writing your report is provided within the lab. However, if you wish to word process your lab report, your instructor will e-mail you a copy of this lab as an attached MS Word documentPLEASE KEEP A COPY OF YOUR COMPLETED LAB REPORT.
You may need to refer to the work you did on this lab before it is graded and returned.

Recall that means the limit of the Riemann sums Sn = . In these sums, n is the number of subintervals into which the interval from –1 to 2 is divided by equally spaced partition points
–1 = x0x1 < … < xn-1 < xn = 2. The length of each subinterval is Δx = = . (The subintervals don’t have to be all the same length, but calculations are easier when they are.) Each ck is a number from the kth subinterval. This limit statement tells us that is the number that can be approximated as closely as anyone could want (though not necessarily exactly) by Riemann sums computed using a partitioning of the interval – 1 x 2 into a large enough number of equal subintervals.

The partition points x0x1x2 < … < xn-1 < xn that divide the interval of integration – 1 x 2 into subintervals are at the heart of the idea of a Riemann sum. But the evaluation points c1, c2, c3, c4, … , cn-1, cn are more prominent in calculating the value of a Riemann sum. The key to learning to use your calculator to evaluate Riemann sums is to learn to calculate with lists of evaluation points.

I.1.We begin by calculating several Riemann sums S6 for the integral based on dividing the interval of integration into 6 subintervals of equal length. On the graph of f(x) = (next page), divide the interval [ -1, 2] on the x-axis into 6 equal subintervals. Mark and label the endpoints of these subintervals x0, x1, x2, x3, x4, x5, and x6. The seven endpoints of these six subintervals are the partition points.

We’ll first calculate a Riemann sum using the left endpoints of the subintervals as the evaluation points. With this choice the evaluation points c1, c2, c3, c4, c5, c6 coincide with the first six partition points
x0, x1, x2, x3, x4, x5.

Step 1: Read your calculator manual to find out how to create a list. Enter the list { – 1, – 0.5, 0, 0.5, 1, 1.5 } of left endpoints of the subintervals from above into your calculator. Store this list under a name that suggests
Left Endpoints. (I suggestLFTND or lftnd.)

Step 2: To compute a Riemann sum S6 we need to evaluate the integrand f(x) = at each evaluation point in the listLLFTND. TI® calculators will create a new list by evaluating a function at each entry of a specified list. Enter the integrand f(x) = as Y1 on the Y= screen of your TI-83® or TI-84® (or as y1 on the GRAPH screen of your TI-86® or TI89®). Return to the home screen and enter Y1(LLFTND). On a
TI-83® or TI-84®, to enter LLFTND, press [2nd] [list], scroll to LLFTND and press [ENTER]. On a TI-89®, to enter lftnd, press [2nd] [var-link], scroll to lftnd and press [ENTER]. Press [ENTER] a second time. A list of the values of the integrand at the six evaluation points in the list LLFTND will be displayed. You may want to store this list under a name that suggests Function Values. (I suggestFNVLS or fnvls.)
The numbers in this list are the successive heights of the 6 rectangles that approximate the area under the graph and, hence, the integral. Draw these approximating rectangles on the graph on the next page.

Step 3: To get the areas of the approximating rectangles, we must multiply each of the function values in the list LFNVLS (or fnvls) you just created by the length of the bases of the approximating rectangles. The length of the base of each rectangle is Δx = = 0.5. Multiplying a list by a number multiplies each entry of the list by that number. To find the area of each of the approximating rectangles in the Riemann sum, multiply 0.5 times the list of function values. (On a TI-83®, TI-84®, or TI-89® enter 0.5* [2nd ], (ANS) and press [ENTER].)

Step 4: The value of the Riemann sum S6 calculated with left endpoints as evaluation points is the sum of the six entries in the list of areas you just created. The SUM command adds all the entries in a list.
(To find the SUM command on a TI-83® or TI-84®, press [2nd ], (LIST), scroll to MATH, and chooseSUM from the menu. On a TI-89®, press [2nd ], (MATH), scroll to LIST, press the right arrow, and chooseSUM from the menu.) Apply the SUM command to the list of areas of rectangles (use sum(2ndANS) ) and verify that S6(left endpoint evaluations) = 3.250969074 (from a TI-84®).

I.2.The approximation for calculated in I.1 is not very accurate.

(a)Use the graph of f(x) = above to explain why S6 calculated with left endpoint evaluations is smaller than the actual value of .

(b)Without evaluating the integral or finding the actual area, estimate the amount (not the percentage) by which S6 calculated with left endpoint evaluations differs from the actual value of. Explain how you arrived at this estimate. (Method for estimating error must be plausible. Explanation must be consistent with method used to estimate the error.)

(c)Riemann sumapproximations for calculated using left endpoints as evaluation points will always be smaller than the actual value of the integral. What property of the function accounts for this?

Use agraph and words to explain why it is that if a function y = f(x) has the property you just stated, then every Riemann sum approximation for calculated using left endpoints as evaluation points will be smaller than the actual value of the integral.

II.1.Next, calculate the Riemann sum S6 for the integral using the right endpoints of the subintervals as evaluation points. To get the right endpoints of the subintervals, add the length of the subintervals (Δx= 0.5) to the left endpoints. To recall the list of left endpoints { -1, -0.5, 0, 0.5, 1, 1.5 } from memory press [2nd], (LIST), scroll to the name you gave the list of evaluation points, and press [ENTER]. Adding a number to a list adds the number to each entry of the list, so just add 0.5 to this list. (Enter 0.5+ [2nd ] (ANS), [ENTER].) Store this new list under a name that suggests Right Endpoints. (I suggestRTNDor rtnd.)

Follow Steps 2 - 4 from #1 to compute S6 using right endpoint evaluations. Illustrate (label!) and interpret (explain!) all steps of this calculation on the graph of f(x) = below.

S6(right endpoint evaluations) = ______(to full accuracy of your calculator)

II.2The approximation for using right endpoint evaluations is not very accurate.

(a)Use the graph of f(x) = above to explain why S6 calculated with right endpoint evaluations is larger than the actual value of .

(b)Without evaluating the integral or finding the actual area, estimate the amount (not the percentage) by which S6 calculated with right endpoint evaluations differs from .
Explain how you arrived at this estimate.

(c)Riemann sum approximations for calculated using right endpoints as evaluation points will alwaysbe larger than the actual value of the integral.What property of the function accounts for this?

Use a graph and words to explain why it is that if a function y = f(x) has the property you just stated, then every Riemann sum approximation for calculated using right endpoints as evaluation points will be larger than the actual value of the integral.

III.1.Calculate S6 using mid-point evaluations. Illustrate and interpret all steps of the calculations on the graph of
f(x) = below.

S6(midpoint evaluations) = ______(to full accuracy of your calculator)

III.2.Do you think the approximation to the integral calculated using midpoint evaluations is larger or smaller than the actual value of the integral? Explainwithout finding or using the exact value of the integral why you think so.

IV.1.(a)Most calculators will evaluate integrals numerically. Use the manual for your calculator to learn how to evaluate integrals numerically. Evaluate and record the result. ______

(b)Which of the approximations to the integral calculated above is closest to the value your calculator gives for the integral? Explain why one might expect that.

(c)If possible, draw the graph of a continuous function y = f(x) on the interval [ –1, 2] and a partition of the interval [ –1, 2] so that the Riemann sum calculated using left endpoint as evaluation points gives a better approximation for than the Riemann sum calculated using midpoints as evaluation points. If it is not possible to graph such a function and partition, explain how you know.

Page total ______