Defining and Computing Definite Integrals
Overview: In this lesson, students will learn what a Riemann sum is and be given a stepby-step procedure of how to formulate them. They will also learn how to calculate both
upper and lower Riemann sums. They will examine Riemann sums in depth through
geometer’s sketchpad. A formal definition of a definite integral will be discussed and
students will learn and use integral notation – integrand, limits of integration, variable of
integration, and what it means for an integral to be Riemann integrable. The students will
also discover the properties of definite integrals and explore how to use graphs to
calculate definite integrals over intervals. Finally, they will learn how to calculate the
worst possible error of an approximation integral.
Grade Level/Subject: This lesson is for 12th graders in AP Calculus.
Time: 3-50 minute class periods
Purpose: The purpose of this lesson is to define definite integrals using Riemann sums.
By doing this, students will truly understand how integrals work rather than just learning
a formula. They will be able to make very good approximations using Riemann sums
and compare their approximations to the real answers in order to calculate their error.
They will also learn many properties of the definite integral which will help them to
perform their integrations faster.
Prerequisite Knowledge:
Students should:
- Understand the summation notation
- Understand the basics of Geometer’s Sketchpad
Objectives:
1. Students will learn how integrals work through Riemann Sums, instead of just
memorizing a formula.
2. Students will be able to calculate Upper and Lower Riemann sums and the largest
possible error in these approximations.
3. Students will be able to understand and use integral notation-integrand, limits of
integration, and variable of integration.
4. Students will understand and be able to use the properties of definite integrals
5. Students will learn how to use graphs to calculate definite integrals over intervals.
Standards:
1. Communications: Students will be asked to communicate what they learn through
homework and through the Geometer’s Sketchpad activity.
2. Problem-Solving: Students will use their problem solving skills in order to
calculate upper and lower Riemann Sums and to approximate area using different
geometric shapes.
3. Technology: Students will use Geometer’s Sketchpad to explore Riemann Sums.
Resources/Materials Needed:
1. Computer Lab with Geometer’s Sketchpad
2. Handouts for Geometer’s Sketchpad Lab
3. Calculus Book
4. Paper/Pencil
Activities and Procedures:
1. Introduce Riemann sums. Draw a function on the board and draw in the
rectangles to give the students a graphical representation of what a Riemann Sum
is.
2. Go through the steps of producing a Riemann Sum.
Step 1: For some positive integer n, subdivide the interval [a,b] into n subintervals
of equal length. This forms what is called regular partitions of [a,b] of size n.
ba
Each subinterval length, x 
.
n
If we label our starting point as a = a0. The partition points would be:
a0 = a, a1 = a + x , a2 = a +2 x , a3 = a + 3 x , …, an = b = a + n x
Step 2: Choose one point xi (i = 1, 2,…,n) from each of the n subintervals. Each
endpoint may be chosen from anywhere in the subinterval, including the
endpoints. Common choices are the left or right endpoint, or the midpoint of each
subinterval.
Step 3: Calculate f(xi) for each xi chosen, multiply each function output by x ,
and sum the results:
f ( x1 )  x  f ( x2 )  x  f ( x3 )  x  ...  f ( xn )  x
This is the value of the Riemann sum for f over [a, b] for this partition and choice
of points.
3. Have the students calculate the Riemann sum of f(x) = x2 over the interval [0,3]
with a regular partition of n = 4 subintervals and with chosen points x1 = 0.5, x2 =
1.0, x3 = 2.0, x4 = 2.7.
4. Explain how summation notation can be used for donating Riemann sums.
5. Introduce Upper and Lower Riemann Sums. A lower Riemann sum corresponds
to the sum of the areas of the inscribed rectangle. An upper Riemann sum
corresponds to the sum of the areas of the circumscribed rectangle. If a
continuous function is monotonically increasing or decreasing, the minimum and
maximum output values will always occur at the endpoints of each subinterval.
6. Have each group find the upper and lower Riemann sums for f(x) = x2 over the
interval [0, 3] with a regular partition of n = 4 subintervals.
Day one Homework: Read section 6.2 and complete the attached worksheet #1
Day 2:
7. Go over Worksheet from previous Day
8. Riemann Sum Activity Day: Go to computer lab and let students explore Riemann
sums through Geometer’s Sketchpad. The Lab worksheet is due at the beginning
of class the following day.
Day 3:
9. Go over Lab Activity from previous day.
10. Give the formal definition of a definite integral. Let f be a function defined on a
closed interval [a, b]. If the Riemann sums for f over [a, b] converge on a single
limiting value A as the partition gets finer and finer then we write
n
b
i 1
a
A  lim  f ( xi )x   f ( x)dx and we say that f is Riemann integrable over the
n
interval [a, b]. Here f(x) is called the integrand, and a and b are the limits of
integration, and dx denotes the variable of integration.
11. Have each student compute the following definite integral. Approximate the
1
1
upper and lower Riemann sums for  x dx using a regular partition of size n = 5.
2 2
Once they have these two values, explain how to average the upper and lower
approximations. Sometimes this average will be a better approximation,
sometimes it won’t be.
12. Discuss how to find the largest possible error.
U  Ln
Largest possible error = n
. Calculate this error from the previous example.
2
13. Properties of Definite Integrals:
b
a. Additivity over Intervals: If a<b<c, then

c
c
f ( x)dx   f ( x)dx   f ( x)dx
a
b
a
x2
dx
3
5
4
Draw a graph to demonstrate this property. Do an example: Find

b
b
a
a
b. Constant Multiple Property: If c is constant, then  cf ( x)dx  c  f ( x)dx
Prove this by showing that you can factor a constant out of the Riemann sum.
b
b
b
a
a
a
c. The Additive Function Property:  ( f ( x)  g ( x)) dx   f ( x)dx   g ( x)dx
Prove this using Riemann sums as well.
d. Comparison Property: If f(x)  g(x) for every x  [a, b], then
b

a
b
f ( x)dx   g ( x)dx
a
14. Talk about finding a definite integral over a function that is not continuous. It
depends on how many discontinuities there are and how bad they are. If there are
only finitely many and they are removable, then we can find a definite integral.
4
15. Give an example of how this works. Find
 f ( x)dx where
1
f(x) = { 1,  < x <-2, 2, -2  x  1 , -1, 1 < x  3, and 1, 3 < x <  }.
Day 3 Homework: Complete attached Worksheet #2, Read Section 6.3 and take notes.
Name: __________________
For each given interval [a, b], find x and the partition points a0, a1, …, an for a regular partition
of the specified size n. Then indicate the choice of points x1,x2,…,xn corresponding to:
a) left endpoints of each subinterval
b) right endpoints of each subinterval
c) midpoints of each subinterval
1) [a, b] = [-2, 10]; partition of size n = 6.
2) [a, b] = [-3, 4]; partition of size n = 5
3) [a, b] = [7, 12]; partition of size n = 8
Using a partition size of n=5, find the lower and upper Riemann sums for the definite integrals.
1
4)

1  x 2 dx 
0
1
5)
 arctan( x)dx 
1
0
6)  3 x dx 
1
4
7)  log 2 ( x)dx 
1
Use the information below to find the values of the integrals.
b

c
f ( x)dx  2.5
a

d
 f ( x)dx  1.5
f ( x)dx  5.0
b
c
b
8)

d
f ( x)dx
a
9)
 f ( x)dx
b
c
10)

a
d
f ( x)dx
11)
 f ( x)dx
a
Name
________________________
To explore Riemann Sums, we’ll be using Geometer’s Sketchpad. To begin this activity, load
Geometer’s Sketchpad version 4. There are two explorations and each, with their directions, is
given below.
“riemann”
1. After getting in to Geometer’s Sketchpad, open the file called “riemann”. In its initial
setting , you will see one giant circumscribed rectangle (n = 1) approximating the area
between the curve y = -0.61x4+-1.14 x3 +1.75x2 +2.71x+2.46 and the x-axis from x = 1.993 to x = 2.005. We’ll call this the interval [-2,2]. The actual area of 11.322 square
units shows as the solution to the integral on the screen along with the area of the
rectangle(s), N (the number of subintervals), and 4/N (the subinterval width). Click on
Left Rectangles and record what you see on the table in the table below. Click on Left
Rectangles again to unselect it. Then do the same for each of the other Reimann sums –
right rectangles, midpoint rectangles and trapezoids, recording it in the table below.
Number of
Subintervals
Width of
each
subinterval
Actual Area = _______________
Area Approximation
Left
Right
Midpoint
Trapezoids
Rectangles
Rectangles
Rectangles
Method with least
Error . Write Left,
Right, Midpoint or
Trapezoid
1
2
3
4
5
6
2. Continue changing n until you have explored the partitions from 1 to 6. You should see
the Riemann sum become closer and closer to the desired area and the error decrease.
Record these values in the table above.
3. Did you notice the
button and the
button? Try them.
n+1 nonstop
n+10 nonstop
What do they do?
Let’s try something more versatile. The a, b, c, and d sliders can be used to generate any
polynomial up to a quartic.
1. Change the interval under consideration by sliding X0 and Xn along the x-axis (watch the
lower left information box when selecting so as not to grab other points which are
concurrent with Xn.
Set your sliders to the function below. Leave n set to six subintervals (partitions).
Choose sums generated by left endpoints, right endpoints, midpoints, or trapezoids by
double clicking the appropriate button in the upper left of the sketch. You may overlay
more than one choice or see one at a time by double clicking “Hide” for the others. Fill
out the box below for six subintervals:
Function:
Set a new interval: Drag (Capital A on the graph) A (x0) to –0.979 and leave B (x6) at
2.005 - this is as close as we can get to the interval [-1, 2] Use [-1,2] for x calculations.
Left
Right
Midpoint
Trapezoid
X0
X6
n x Endpoint
Endpoint
Endpoint
Endpoint
Actual
Approx.
Approx.
Approx.
Approximation Area
6
2. Note: You can re-scale the axes by dragging the “Unit Point” – its label is not showing
but it resides at position (1, 0). Move the scale so that you can see the area UNDER the
x-axis clearly.
Change the interval under consideration by sliding X0 to –1.001 (as close as you can get)
and X6 to 0.000 (as close as you can get) along the x-axis (watch the lower left
information box when selecting so as not to grab other points which are concurrent with
Xn. Use [-1,0 ] for x calculations.
X0
X5
n
x
Left
Endpoint
Approx.
Right
Endpoint
Approx.
Midpoint
Endpoint
Approx.
Trapezoid
Endpoint
Approximation
Actual
Area
5
3. A) Change your function by using the sliders to graph a new function
B) Change n to set the desired number of subintervals from 1 to 6.
C) The exact value of the integral is expressed as an interval. Complete the interval and
write the actual area in the interval below. Finish filling out the table below.
Function:_f(x) = 2.00 x4 – 3.00 x3 – 1.00 x2 +2.50 x + 0.00
X0 = –1.001
n
x
Left
Endpoint
Approx.
Xn = 1.001
Right
Endpoint
Approx.
Actual area: = _____________
Midpoint
Endpoint
Approx.
Trapezoid
Endpoint
Approximation
Best estimate given by ____
method? (write LEFT,
RIGHT, MID, or TRAP)
1
2
3
4
5
6
4. On a separate paper, write a paragraph for each of these ( 10 points each)
a) From the information you have collected and things that you did with Riemann sums,
write some things that you now know about the area under a curve. Make sure to address
things that you found out in #2 at the top of this page.
b) What ways do you have to compute approximations of areas under curves? Is one way
better than others? Why or why not?
Name: ___________________________
Consider the definite integral:
4
 x( x  1)( x  2)( x  3)( x  4)dx
0
1) Find the Riemann sum corresponding to the regular partitions of size n = 1, 2, 4,
8, using the left endpoint of each subinterval.
2) Find the Riemann sum corresponding to the regular partitions of size n = 1, 2, 4,
and 8, using the right endpoint of each subinterval.
3) Find the Riemann sum corresponding to the regular partitions of size n = 1, 2, 4,
and 8, using the midpoint of each subinterval.
4) Graph y = x(x-1)(x-2)(x-3)(x-4) over the interval [0, 4] and use the graph to
explain the results of exercises 1-3.
3
5) Compute the definite integral
 x dx as a limit of Riemann sums.
2
0
5
6) Find

2
x3 4
2
dx