Numerical Integration Techniques
Overview: In this lesson, students will be introduced to five different numerical
integration techniques. They will be asked to use each different method and to decipher
what particular method is more accurate for particular problems.
Grade Level/Subject: The lesson is for 12th graders in AP Calculus.
Time: 1-50 minute class period
Purpose: Even though the students have just learned the 2nd Fundamental Theorem of
Calculus for computing definite integrals, this section provides them with five other
techniques that can be used. Numerical integration is important for when an
antiderivative cannot be found so that the 2nd Fundamental Theorem cannot be used.
Prerequisite Knowledge:
Student should:
- Understand Definite Integrals
- Understand Riemann Sums
Objectives:
1. Students will learn how to use each of the five numerical integration techniques
comfortably.
2. Students will learn which method is more accurate for particular problems.
Standards:
1. Problem-Solving: Students will use their problem-solving techniques to determine
what method is more accurate for particular problems.
2. Technology: Students will be given a program for their calculators that can
perform all five of the numerical integration techniques.
3. Communications: Students will have to communicate clearly what method is
being used and all the steps taken to achieve their answer.
Resources/Materials Needed:
1. Calculus Book
2. Dry Erase Board/Dry Erase Markers
3. Numerical Integration Program for TI-89
Activities and Procedures:
b
1. Begin by reminding the students that a definite integral
 f ( x)dx
is defined to be
a
n
the limiting value of Riemann sums lim
n 
 f ( x ) x
i 1
i
i
where the interval [a, b] is
partitioned into subintervals of length x , one input xi is chosen from each
subinterval, and we sum up the products f(xi) x for all the subintervals.
2. Introduce the Left endpoint, Right endpoint, and midpoint rules. Explain that the
difference between the three rules is simply the choice of input made for each
subinterval.
Step 1: Choose a number n of subintervals in the regular partition of [a, b].
ba
Step 2: Calculate x =
.
n
Step 3: Locate the n inputs, x1, x2,…, xn. (This is where the difference comes in!)
Step 4: Evaluate f at each input xi and find the Riemann sum:
n
 f ( x )x  f ( x )x  f ( x
i 1
i
1
2
)x  f ( x3 )x      f ( x n )x )
3. For the left endpoint rule, the inputs will be:
x1 = a, x2 = a + x , x3 = a + 2 x ,… ,xn = a + (n-1) x
4. For the right endpoint rule, our inputs will be:
x1 = a + x , x2 = a +2 x , x3 = a + 3 x ,… ,xn = a + n x
5. For the midpoint rule, our inputs will be”
x
3x
5x
(2n  1)x
x1 = a +
, x2 = a +
, x3 = a +
,… ,xn = a +
2
2
2
2
Draw the graph depicting this one as well, just like previous two graphs.
6. Do an example using each of the three rules for a regular partition of n = 3.
3.5
 sin
3
( x)dx . Do this example by hand and then have them do it with the new
0.5
RSUM program on their calculators and compare their results. Also compare
their results to the answer using the 2nd fundamental theorem to see which method
was most accurate.
7. Introduce the trapezoidal rule.
Trapezoidal rule estimate =
1
1
(Left Rectangle estimate) + (Right Rectangle estimate).
2
2
8. Have them use the trapezoidal rule to solve the previous example among their
groups by hand and on their calculators. Compare the class’s answers. Also
discuss if the trapezoidal rule was more accurate or not.
9. Introduce Simpson’s Rule. Simpson’s Rule is the weighted average of the
midpoint and trapezoidal rules. The motivation for this is to account for the
concavity of the function’s graph over each subinterval.
Simpson’s rule estimate =
1
2
(Trapezoidal estimate) + (Midpoint estimate)
3
3
10. Have them use Simpson’s rule to solve the previous example among their groups
by hand and using their calculators. Compare the class’s answers once again.
Discuss whether or not this method was more accurate.
11. Summarize all five techniques and the notation used by each one.
Homework: Read Section 6.7 and take notes. Complete the attached worksheet.
Name: __________________
Using regular partitions of size n = 2, 4, 8, 16, and 32, find the left, right, and midpoint
estimates for the definite integrals. (Do 2 and 4 partitions by hand and 8, 16, and 32
partitions using your calculator.)
1
1)

1  x 2 dx
0
1
2)
 arctan( x)dx
1
1
3)  e  x dx
2
0
Using regular partitions of size n = 2, 4, 8, 16, and 32 and the results from the previous
exercises, calculate each integral using the trapezoid rule and Simpson’s rule. (Do 2 and
4 partitions by hand and 8, 16, and 32 partitions using your calculator).
1
4)

1  x 2 dx
0
1
5)
 arctan( x)dx
1
1
6)  e  x dx
2
0