Math 2015
Lesson 3
Estimating Integrals: Trapezoidal and Simpson’s Rule
We have defined the definite integral as the limit of left and right hand sums, and we
have estimated integrals using left and right hand sums. Unfortunately, it may take a lot
of subintervals to get a good estimate. We have successfully used geometry to help us
calculate a few integrals exactly. (Remember that for f(x) ≥ 0, the integral represents the
area trapped between the curve and the x-axis.) This geometric interpretation will help us
find better estimates to integrals by using shapes other than rectangles to approximate the
area under a curve.
Trapezoidal Rule
Below, we see an approximation to the area under a curve between a and b using just one
subinterval and either a left hand or right hand sum. It’s clear that the rectangles do a
very poor job:
The areas in gray, given by the formulas for the left and right rectangle sums are,
respectively, smaller and larger than the true areas under the curve by more than half the
true area, a relative error of more than 50% in each case. It turns out that improving the
accuracy by dividing a into subintervals and filling in with smaller rectangles, as done in
Section 5.2, is less efficient than first improving the fit for one basic region and then
subdividing. The simplest such method is the trapezoidal rule, in which the area under
the curve between a and b is approximated by a trapezoid, as in the following diagram:
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Math 2015
Lesson 3
The area of a trapezoid is the average height times the width, or in this case:
Here, we are using one subinterval, so b – a is really just ∆x. If we want to improve our
answers even further, we can subdivide the interval into n subintervals of width ∆x = (b –
a)/n, and add up the result from the trapezoidal rule on each subinterval. We will label
the division points with x0 = a, x1 = a + ∆x, …, xn = b, just as with writing down the left
and right hand sums:
Then the trapezoidal rule might be written down as
This is a little bit of a mess, so let’s simplify. Each term has a ∆x/2 term, so let’s factor
that out to the front:
In fact, we can do better than this! Note that each of the middle points is used twice: as
the left side of one interval and the right side of another. So we don’t need to compute
these separately. We finally get the following calculation for the Trapezoidal rule:
Example:
Approximate
∫
4
0
x 2 dx using the Trapezoidal rule with n = 8 subintervals.
First, we should figure out Δ x and set up our table of values.
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Math 2015
Lesson 3
It’s worth asking how this compares to the left and right hand sums for
∫
4
0
x 2 dx . The left
hand sum with n = 8 is
while the right hand sum is
The average of the left and right hand sums is _______, which is exactly the same as the
trapezoidal approximation!
This actually shouldn’t be a surprise. After all, the area of each trapezoid is just the
average of the areas of the left and right hand rectangles. So what we have really found
is a shortcut formula to allow us to calculate this better estimate without first calculating a
left and right hand sum. (We also have a good reason to believe it is a better estimate,
based on our area argument!)
64
≈ 21.3 , so the trapezoidal estimate is
0
3
pretty good here. (We will learn how to determine the exact value of integrals like this
later in the semester.) We can see in the picture below, with trapezoids superimposed on
the graph, that the approximation ought to be close; it’s hard to even tell the difference
between the trapezoids and the curve itself.
It turns out that the actual value is
∫
4
x 2 dx =
15
12.5
10
7.5
5
2.5
1
2
Trapezoidal rule with n = 8 for
3
∫
4
4
0
x 2 dx
When we are approximating a function using the
trapezoidal rule, we are essentially replacing the
function with a piecewise linear function and
integrating that, as in the picture at right.
The problem is that the function may not be very
well approximated by a piecewise linear
function. It’s clear we could do better if we
worked with higher degree curves, such as
parabolas.
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Math 2015
Lesson 3
Simpson’s Rule
Using higher degree curves to approximate our function leads to a second approximation
technique, which is known as Simpson’s rule. To use Simpson’s rule, we choose n and
calculate ∆x as before, and find the same set of points x0 = a, x1 = a + ∆x, …, xn = b. We
then use the following formula:
Simpson’s rule:
This formula looks a lot like the trapezoidal rule, except for two things:
•
∆x is divided by
•
the coefficients for the “middle” terms alternate between and
ending with a
in the second and next-to-last term.
, and
, starting and
It is important to note: Simpson’s rule must be used with an even number of
subintervals. (This is because each parabola is actually stretched across a pair of
subintervals, not a single subinterval.) Thankfully, you will get a hint if you’ve forgotten
and used an odd number of subintervals: if you start with coefficient 4 in the second
place, the next to the last place must have coefficient 4. (If n were odd, you would end
up with a 2 instead.)
We will not give a derivation of this formula as we did for the trapezoidal rule, partly
because we are not familiar with the formula for the area under a parabola.
Example:
Let’s get a new estimate for
∫
4
0
x 2 dx using Simpson’s rule with n = 8
subintervals. So, ∆x = ______ . And we can use our same table.
Now do the same problem with n = 4 subintervals.
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Math 2015
Lesson 3
Do you notice something interesting about your answer compared to the actual answer
we gave earlier? Why would this be true?
Tables of Data
Sometimes we may work with tables of data rather than functions described
algebraically. As long as the data is evenly spaced, we can still use Simpson’s or the
trapezoidal rule to estimate the integral. (We may often have to use the trapezoidal rule,
because there may not be an even number of subintervals.) For data which is not evenly
spaced, we could still average the left and right hand sums.
Example:
Suppose the following data is known for a function W(t). Give the best
∫
estimate you can for
t
W(t)
4
−2
W (t) dt .
–4
–2
0
2
4
7
4
3
-1
2
HINT: Be sure you know exactly what data you will be using!
Here, ∆ t = _______.
NOTE: When using these rules with tables of data, your ∆x will be determined from the
given information (in some way). It is not necessary to use our (b – a)/n formula.
Example:
To estimate the surface area of a pond, a surveyor takes measurements
every 20 ft. across the width of the pond. The following data is collected:
measurement #
1
2
3
4
5
6
7
8
9
width in feet
0
50
54
82
82
73
75
80
0
What is the approximate total surface area?
Before we begin, read the problem carefully and determine what ∆x will
be in this case:
∆x =
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Math 2015
Lesson 3
Next, we must determine an appropriate method, and do the calculations:
Follow-up Question: Suppose the pond above has an average depth of 10 feet, and we
plan to start fishing season with one fish per 1,000 cubic feet of water, how many fish
need to be in the pond on opening day? (Hint: Start by finding the volume of the pond.)
Summary
Today, we have
•
Developed a formula for the trapezoidal rule, which approximates integrals by
finding the area between the curve and axis using trapezoids. (The trapezoidal
rule replaces the function with straight line segments, and integrates this simpler
curve.)
•
Used a formula to estimate an integral using Simpson’s rule, which uses parabolas
to approximate the curve. (Therefore, Simpson’s rule gives the exact value of the
integral of a polynomial of degree two or lower.)
•
Determined that the approximations generally get better as ∆x gets smaller.
•
Determined that Simpson’s rule generally gives better approximations than the
trapezoidal rule, which is in turn better than the left or right hand sums.
•
Used the trapezoidal and Simpson’s rule to approximate integrals from tables of
values for a function.
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