Maple Project 3 for Sections 3 and 8
In this project you will explore how an integral is a limit of Riemann sums by considering
a specific example, namely, the integral from 0 to 1 of the function f(x) = cos(x2).
1. Have Maple make a plot of the function showing 100 rectangles corresponding to
the left Riemann sum with n = 10. Use the leftbox command described below for
this. Then have Maple calculate the value of the left Riemann sum with n = 10
using the leftsum command. Do the same thing (make a plot and find the value)
for the right Riemann sum with n = 10.
2. By how much to the left and right sums you found in problem 1 differ? If you
averaged them and used the average as an estimate of the integral how far off
could you be? Explain.
3. Find the values of the left Riemann sums for n = 10, 100, 1000, 10000.
4. Suppose you want to find the value of the integral to within .001 by using a left
Riemann sum. How small a value of n (not necessarily one of the values of n
you found in question 3) could you use for this? Use Maple to experiment and
report your findings. Assume that you have no other way of determining the
exact value of the integral.
5. * Verify that your answer in 4 is correct by computing the corresponding right
Riemann sum, along with the left Riemann sum, for the value of n you proposed
in problem 4.
INSTRUCTIONS
Open the template worksheet “maple3sec3-8.mw”. This worksheet will have a place for
you name, which must be entered in the worksheet, an executable line
restart; with(plots): with(student):
that you should execute by placing the cursor on the line and hitting Enter,
and the Maple definition of the function f(x) = cos(x2). If this worksheet does not open
in Maple when you click on the link you may proceed in two ways
i) When you see what looks like gibberish on the screen, have the browser save it to
the desktop. It will automatically have the right name. Then exit the browser
and click on the resulting icon. The worksheet should now open in Maple.
ii) Click on the link to the “readable” version of the template worksheet and copy
what you see into a new Maple worksheet.
Answer questions 1. – 5. in the worksheet, using COMPLETE SENTENCES if the
problem calls for a non-graphical answer.
You must submit your work in printed form, no electronic submission without explicit
permission.
Maple homework should have only one author. You may discuss the
project with your classmates, but what you turn in should contain your own answers.
Plagiarism is a serious academic offense. Insert text comments to explain your work.
Neatness and good English count. Maple should be used for all calculations and plots.
You will find the commands on the next page useful for this assignment.
USEFUL MAPLE COMMANDS
> f:=x->x^2;
defines the function f(x)=x^2.
> leftbox( f(x), x = 0 .. 1, 20); will produce the graph of f between 0 and 1 showing
20 rectangles corresponding to a left Riemann sum with n = 20
You could also just enter the function in this command, e.g.
> leftbox( sin(x) , x = 0 .. 1, 20); will produce the graph of sin(x) between 0 and 1
showing 20 rectangles corresponding to a left Riemann sum with n = 20
>rightbox( f(x), x = 0 .. 1, 20); does the same as leftbox but for right sums
>evalf( leftsum( f(x) , x = 0 .. 1, 20)); will produce the value of the left sum

with n = 20
>evalf( rightsum( f(x) , x = 0 .. 1, 20)); will produce the value of the right sum

with n = 20