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Riemann
Sums: A Symbolic and Graphical Approach
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Objective: This demo uses a
computer algebra system to investigate various approximations to the definite
integral. Right hand endpoint, left hand endpoint and midpoint Riemann sums as
well as trapezoidal and Simpson's rules are expressed in closed form as
functions of 
. Behavior as
approaches 0 is observed. This demo helps students to
- understand the definition of Riemann sum and
why Riemann sums approximate the definite integral;
- demonstrate that the limit of a Riemann sum is the
same for (almost) any choice of points from each subinterval;
- investigate combinations of Riemann sums that lead to
better approximations to the definite integral;
- appreciate the meaning of rate of convergence of
an approximation.
Level: This demo can be presented
in any course in which definite integration using Riemann sums is discussed. While
this will often be a topic of discussion in a Calculus class, it is also
appropriate for a numerical analysis class.
Prerequisites: Students
should be familiar with Riemann sums. The concepts of upper and lower sums should have been introduced. In
addition, it would be useful if students have been introduced to graphical
notions of limits and limiting behavior. Simpson's rule and the
trapezoidal rule are also illustrated, that portion of the demo should be shown
after students have been introduced to these approximate methods.
Platform: Any computer system
with Mathematica or Maple. The downloadable worksheets were
developed in Mathematica 4 and Maple 6.
Instructor's Notes:
As first examples, it is helpful to investigate common
functions, such as polynomials.
NOTE:
When the definite integral is introduced, it is often an area calculation.
This the case when the function is nonnegative over the interval. The
example in this demo leads to an appreciation that the definite integral is not
always an area of a region and thus is not always positive.
Evaluate 
.
A
usual and important approach is to partition the interval [0,2] into n equal
subintervals and construct n rectangles over the subinterval with (signed)
heights determined, for example, by right hand endpoints, left hand endpoints,
or midpoints of the subintervals. An approximation for the definite
integral is determined by the sum of the signed areas of the n rectangles.
For this example, we construct the Riemann sum for this integral by dividing the interval [0,2] into
n subintervals of equal width so that
= 2/n.
Using right hand endpoints, the Riemann sum
that approximates the definite integral (computed here by Mathematica)
is
=
NOTE:
In this example, the interval is [0,2], so the points at which we evaluate the
function are multiples of .
For a general interval [a,b] and using right hand endpoints, we evaluate
the function at 
, where k = 1, 2,
..., n. This remark extends to the other approximation methods as well.
The
rectangles determined using right hand endpoints, and the computed values for
the Riemann sums for n = 1 to n = 30 are displayed in the animation in Figure
1. The left hand graph shows the rectangles while the right hand side
shows a graph of the values of the Riemann sums as they are computed. As n becomes
larger, the rectangles appear to fill the region between the curve and the
x-axis. It also appears that the approximations seem to be leveling off to some (as yet
unknown) value. Note that for n = 1, the rectangle is completely below the
x-axis, so the approximation is negative. For n > 1, there are
rectangles above and below the x-axis, providing some cancellation.
However, it is reasonable to think there is a strong possibility that if
the approximations are approaching a limiting value, that value will be
negative.
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Figure 1. Riemann sums
computed using right hand endpoints. |
Using left hand endpoints, the Riemann sum is
=
Figure
2 shows the approximating rectangles for the left hand endpoint Riemann sum and
the corresponding approximations to the definite integral.
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Figure 2. Riemann sums
computed using left hand endpoints. |
We will now compute the Riemann sum using
the midpoints of the subintervals. The midpoint, mk, of each
subinterval 
with width
can be computed by averaging the endpoints. Thus, for this approximation
scheme, we evaluate the function at 
.
(On
a general interval [a,b], the midpoints are 
.)
For
this example, the
midpoint approximation is
=
Figure
3 shows the approximating rectangles for the midpoint Riemann sum and the
corresponding approximations to the definite integral.
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Figure 3. Riemann sums
computed using midpoints. |
In each of the previous cases, it appears that the
computed values of the Riemann sums will eventually approach a limiting value as n
becomes large. But does each method
produce the same limiting value? Figure 4 provides convincing evidence
that for this example, the answer appears to be "YES."
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Figure 4. Right
Hand, Left Hand, Midpoint
Approximations |
An
important geometric observation to point out to students is that as n becomes
larger, the width of each rectangle, ,
becomes smaller. This is a nice alternative approach
to the usual textbook presentation because considering
the limit as ->0
is an easier concept for many students to understand than the more abstract idea
of a limit as n -> Infinity. This is also an easier approach from a
computational standpoint, particularly when the Riemann sums for
a polynomial can be computed
exactly using a symbolic algebra package.
The
symbolic expressions we have obtained so far for the Riemann sums involve both n
and .
To express the sums in terms of
only, substitute n = 2/.
The simplified expressions (using Mathematica) for the right hand
endpoint, left hand endpoint, and midpoint Riemann sums are, respectively,
RHE(
) = 
,
LHE()
= 
,
and
MidPt( )
= 
.
From these
symbolic expressions, it is easy to see that as
approaches 0, all three expressions approach the exact limiting value -218/15,
approximately -14.533.
Figure
5 shows plots of the graphs of the right hand, left hand, and midpoint
approximations generated as approaches
0.
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| Figure 5. Right
hand, left hand, and midpoint approximations generated as
approaches 0. (For these calculations, we used n = 50 and
displayed every other frame.) |
From Figure 5, it seems that the midpoint Riemann sums approach the limiting value
faster than the other two approximations. NOTE: This
observation allows us to introduce the idea of "rate
of convergence," discussed below.
This example can convince students that Riemann sums using
left and right
hand endpoints and midpoints of the subintervals converge to the same value as 
,
but what about other partitions of the interval and other choices for the points
in the subintervals at which to evaluate the function? A computer algebra
system cannot be used to compute a Riemann sum in closed form for an arbitrary
partition of the interval, however, if we use a consistent method for choosing
the points in the subintervals having equal widths, we can obtain a closed form
expression for the Riemann sum.
One
way to accomplish this is to require that
= 2/n and choose the points xk* inside each subinterval by
the following
and s varies between 0 and 1. If s = 0, this corresponds
to choosing right hand endpoints and if s = 1, we obtain the left hand
endpoints. If, for example, s = 1/2, the chosen points are the
midpoints of the subintervals.
Using this
scheme, we see that
which
simplifies to
when
we set n = 2/.
It
is clear from this expression that the limit as is
-218/15.
Other Approximation
Schemes: Trapezoidal and Simpson's Rules
Another
approximation we can use involves averaging the approximations for the
right and left hand endpoints. This approximation scheme is called the
trapezoidal rule. If the right and left hand endpoint
approximations are expressed as functions of ,
the trapezoidal rule is
=
.
Another combination of the approximations yields one form
of Simpson's rule:
= 
.
Note: This is not the usual
formulation for Simpson's rule. See the reference to find an explanation of
how this form corresponds to the classical formulation for Simpson's rule.
It is clear from these expressions that each
will approach the limit -218/15 as .
But how do these approximation schemes compare to the
previous schemes? To answer that question, we introduce the idea of "rate of convergence."
Rate of
Convergence
The rate of convergence of a numerical
algorithm for approximating a limiting value gives a general idea of how fast
the algorithm approaches the limit. From a computational point of view,
the faster the algorithm converges, the better. One way to quantify the
rate of convergence is by looking at the absolute value of the errors generated
by the approximations.
Looking at the symbolic
expressions for the
various approximations,
RHE(
) =
LHE()
=
MidPt(
)
=
=
= ,
we see that if
is very close to zero, the absolute value of the errors
in the approximations are dominated by the term that is proportional to the
lowest nonzero power of .
We will refer to this term as the dominant term in the error. The
dominant terms in the error for each of the right hand and left hand Riemann
sums are proportional to (we
say that the convergence rate is linear). For each of the midpoint and
trapezoidal approximations, the dominant terms in the error are proportional to 
(we say that the convergence rate is quadratic). The dominant term in the
error for Simpson's rule is proportional to 
.
Generally, the higher the (nonzero) power of
in the dominant term in the error,
the faster the convergence to the limit. At
this relatively early stage in Calculus, these ideas do not mean much to
students, however, a graphical approach to these ideas can be very
helpful.
A dramatic visual tool for the
rate of convergence is to plot the values of the various approximation methods
as a function of
and examine n, ,
error, and the ratio of errors at consecutive steps, as we halve .
If the dominant term in the error is proportional to ,
the ratios should approach 1/2. If the dominant term in the error is
proportional to ,
the ratios should approach 1/4. Similarly, if the dominant term in the
error is proportional to , the ratios
should approach 1/16. This also provides a useful tool for discussion of
the meaning of "big oh" notation in a numerical analysis
course.
The animations below illustrate
the linear convergence of the right hand and left hand approximations and the
quadratic convergence of the midpoint and trapezoidal approximations. The
rapid convergence of Simpson's
rule due to the fact that dominant term in the error is proportional to . Note that vertical scales are
adjusted so that behavior near
= 0 can be more easily observed. Convergence for the midpoint and
trapezoidal rule approximations is fast, but convergence for Simpson's rule
is much faster. A table of the data generated by each approximation scheme is interesting to include in a discussion after the presentation of the animations.
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Right Hand Endpoint
Approximation
DATA
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Left Hand Endpoint Approximation
DATA
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Midpoint Approximation
DATA
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Trapezoidal Rule Approximation
DATA
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Simpson's Rule Approximation
DATA
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Other interesting examples are cubic polynomials,
trigonometric functions, and the ideas extend quite naturally to double
integrals.
Additional Resources:
Mathematica 4: Click here
to
preview and download a Mathematica notebook that generates the
calculations and animations
illustrated in the demo.
Maple 6: Click here to preview and
download a Maple 6 worksheet that accompanies this demo.
Credits: This demo was adapted from
Lawrence H. Riddle,
Symbolic
and Graphical Investigations of Riemann Sums with a Computer Algebra System,
PRIMUS, VI, No. 4 (December 1996), 366-380.
We
appreciate the demo suggestion and accompanying Maple worksheet from
Dr.
Larry Riddle
Department of Mathematics
Agnes Scott College
Decatur, GA 30030
Additional Mathematica and Maple worksheets were developed by Lila Roberts.
Additional information regarding possible
use of this demo:
In his class, Dr. Riddle provides a Maple
worksheet that allows students to investigate left, right, and midpoint Riemann
sums as well as sums obtained from the trapezoidal and Simpson's
rules. Each sum simplifies to a closed-form formula in terms of the
number of points in the partition. The student can then investigate such
issues as what happens to the sum in the limit as n goes to infinity or how the
errors for the different sums change as n changes. The worksheet also
allows students to plot the approximations as functions of n to illustrate the
convergence rates to the limit. The goal of the worksheet is to help
students to realize that (at least for polynomials)
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All five of the sums converge to the
value of the integral as n -> Infinity.
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The left and right sums approximate the
integral with an error that is proportional to 1/n and each error is
approximately the same, but of opposite sign.
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The midpoint and trapezoidal
approximations have an error that is proportional to 1/n2 and
that the midpoint error is about half the size of the trapezoidal error but
opposite in sign.
The error for Simpson's rule is
proportional to 1/n4.
