In this chapter we’ve spent quite a bit of time on computing
the values of integrals. However, not
all integrals can be computed. A perfect
example is the following definite integral.
We now need to talk a little bit about estimating values of
definite integrals. We will look at
three different methods, although one should already be familiar to you from
your Calculus I days. We will develop
all three methods for estimating
by thinking of the integral as an area problem and using
known shapes to estimate the area under the curve.
Let’s get first develop the methods and then we’ll try to
estimate the integral shown above.
Midpoint Rule
This is the rule that should be somewhat familiar to
you. We will divide the interval into n
subintervals of equal width,
We will denote each of the intervals as follows,
Then for each interval let be the midpoint of the interval. We then sketch in rectangles for each
subinterval with a height of . Here is a graph showing the set up using .
We can easily find the area for each of these rectangles and
so for a general n we get that,
Or, upon factoring out a we get the general Midpoint Rule.
Trapezoid Rule
For this rule we will do the same set up as for the Midpoint
Rule. We will break up the interval into n
subintervals of width,
Then on each subinterval we will approximate the function
with a straight line that is equal to the function values at either endpoint of
the interval. Here is a sketch of this
case for .
Each of these objects is a trapezoid (hence the rule's name…)
and as we can see some of them do a very good job of approximating the actual
area under the curve and others don’t do such a good job.
The area of the trapezoid in the interval is given by,
So, if we use n
subintervals the integral is approximately,
Upon doing a little simplification we arrive at the general
Trapezoid Rule.
Note that all the function evaluations, with the exception
of the first and last, are multiplied by 2.
Simpson’s Rule
This is the final method we’re going to take a look at and
in this case we will again divide up the interval into n
subintervals. However unlike the
previous two methods we need to require that n be even. The reason for
this will be evident in a bit. The width
of each subinterval is,
In the Trapezoid Rule we approximated the curve with a
straight line. For Simpson’s Rule we are
going to approximate the function with a quadratic and we’re going to require
that the quadratic agree with three of the points from our subintervals. Below is a sketch of this using . Each of the approximations is colored
differently so we can see how they actually work.
Notice that each approximation actually covers two of the
subintervals. This is the reason for
requiring n to be even. Some of the approximations look more like a
line than a quadratic, but they really are quadratics. Also note that some of the approximations do
a better job than others. It can be
shown that the area under the approximation on the intervals and is,
If we use n
subintervals the integral is then approximately,
Upon simplifying we arrive at the general Simpson’s Rule.
In this case notice that all the function evaluations at
points with odd subscripts are multiplied by 4 and all the function evaluations
at points with even subscripts (except for the first and last) are multiplied
by 2. If you can remember this, this is a fairly easy rule to remember.
Okay, it’s time to work an example and see how these rules
work.
Example 1 Using
and all three rules to approximate the value
of the following integral.
Solution
First, for reference purposes, Maple gives the following
value for this integral.
In each case the width of the subintervals will be,
and so the subintervals will be,
Let’s go through each of the methods.
Midpoint Rule
Remember that we evaluate at the midpoints of each of the
subintervals here! The Midpoint Rule
has an error of 1.96701523.
Trapezoid Rule
The Trapezoid Rule has an error of 4.19193129
Simpson’s Rule
The Simpson’s Rule has an error of 0.90099869.

None of the estimations in the previous example are all that
good. The best approximation in this
case is from the Simpson’s Rule and yet it still had an error of almost
1. To get a better estimation we would
need to use a larger n. So, for completeness sake here are the
estimates for some larger value of n.

Midpoint

Trapezoid

Simpson’s

n

Approx.

Error

Approx.

Error

Approx.

Error

8

15.9056767

0.5469511

17.5650858

1.1124580

16.5385947

0.0859669

16

16.3118539

0.1407739

16.7353812

0.2827535

16.4588131

0.0061853

32

16.4171709

0.0354568

16.5236176

0.0709898

16.4530297

0.0004019

64

16.4437469

0.0088809

16.4703942

0.0177665

16.4526531

0.0000254

128

16.4504065

0.0022212

16.4570706

0.0044428

16.4526294

0.0000016

In this case we were able to determine the error for each
estimate because we could get our hands on the exact value. Often this won’t be the case and so we’d next
like to look at error bounds for each estimate.
These bounds will give the largest possible error in the
estimate, but it should also be pointed out that the actual error may be
significantly smaller than the bound.
The bound is only there so we can say that we know the actual error will
be less than the bound.
So, suppose that and for then if E_{M},
E_{T}, and E_{S} are the actual errors for
the Midpoint, Trapezoid and Simpson’s Rule we have the following bounds,
Example 2 Determine
the error bounds for the estimations in the last example.
Solution
We already know that ,
,
and so we just need to compute K (the largest value of the second
derivative) and M (the largest
value of the fourth derivative). This
means that we’ll need the second and fourth derivative of f(x).
Here is a graph of the second derivative.
Here is a graph of the fourth derivative.
So, from these graphs it’s clear that the largest value of
both of these are at . So,
We rounded to make the computations simpler.
Here are the bounds for each rule.
In each case we can see that the errors are significantly
smaller than the actual bounds.
