Table of Contents:
Introduction… 2
Left, Right Mid-Point Riemann Sum Analytical Example… 2
Trapezoidal Sum Analytical Example… 4
AP Level Multiple Choice… 5
Over and Under Approximations… 6
AP Level Conceptual Example… 7
Example Utilizing A Graphing Utility… 8
AP level Free Response Question… 10
Bernhard Riemann… 12
Real-World Application… 12
Works Cited… 13
Analytical Problems… 14
AP Multiple Choice Problems… 15
AP Free Response Question… 16
*All cartoon images and references used in this project are the property of Nickelodeon.
Riemann Sums
Think of a curve on a graph like a ramp. Otto from Rocket Power wants to figure out
the area under the curve or ramp. Why you ask? Because Otto realizes the connection
between skating and math. But how can he attempt to do this?
Riemann sums use rectangles to approximate the area under the curve. The width or
base of the rectangle depends on the interval chosen and it can vary from rectangle
to rectangle. The height of the rectangle depends on the kind of Riemann sum used. A left hand
Riemann sum uses the y-value of the left most coordinate of the interval as the height. A right hand
Riemann sum used the y- value of the right most coordinate of the interval as the height. Finally, a mid
point Riemann sum uses the y-value of the mid-point of the interval as the height.
For example on the function f on the interval (3,5) a left hand Riemann sum will use f(3) as the height of
the rectangle, a right hand Riemann sum will uses f(5) and a mid-point Riemann sum will use f(4).
Finally, the base would be 2 since 5-3=2. To find the area of the rectangle, the basic formula for area
below is used:
The area of the multiple rectangles are then added up together to approximate
the area under the curve. As more rectangles of a smaller width are used, the
approximate becomes more accurate. Think of it like Otto skating down steps.
The larger number and smaller size of the steps, the smoother the ride will be,
and the more like an actual curve it will be.
To obtain the values of f that must be used for the height, you can either use the graph, or sometimes, a
table will be given as will be seen in the example below. A table is useful because it helps organize all the
data that must be used, and it also sometimes reveals what intervals are suggested to be used.
Basic Riemann Sums Analytical Example
Below is a table of the graph f. f(x) is a curve that represents a skateboarding ramp the Rocket Power
crew is trying to design. The problem is Otto’s father said he will only buy
them the wood if they can tell him the area of the wide of the ramp. The x
values of the graph represent the width of the ramp while f(x) represents the
height of the ramp at the width designated by x. Help them approximate the
area by performing a left, right, and mid-point Riemann Sum of three equal
sub intervals!
x
f(x)
0
0
1
1
2
4
3
9
4
16
5
25
6
36
Left Riemann Sum
(0,2)
(2,4)
(4,6)
`
First determine the three equal intervals
(2-0)f(0)+(4-2)f(2)+(6-4)f(4)
Multiply the width of the rectangles by the height
(2)(0)+
Substitute the left most value of the interval in
(2)(4)+ (2)(16)
40
Simplify
Right Riemann Sum
(0,2)
(2,4)
(4,6)
`
First determine the three equal intervals
(2-0)f(2)+(4-2)f(4)+(6-4)f(6)
Multiply the width of the rectangles by the height
(2)(4)+
Substitute the right most value of the interval in
(2)(16)+ (2)(36)
112
Simplify
Mid-point Riemann Sum
(0,2)
(2,4)
(4,6)
`
First determine the three equal intervals
(2-0)f(1)+(4-2)f(3)+(6-4)f(5)
Multiply the width of the rectangles by the height
(2)(1)+
Substitute the mid point value of the interval in
(2)(9)+ (2)(25)
70
Simplify
Applying Integration! You just approximated the integral of the equation given from x=0 to x=6. In other
words, you calculated an approximation of what the area below the curve is! How awesome! Below is
the notation of what you just approximated. Who knew calculus was so easy!
Trapezoidal Sums- Basic Analytical Example (with Buttercup)
Another tool that can help one to find the area under a curve is a trapezoidal sum. A trapezoidal sum is
similar to a Riemann sum but uses trapezoids instead of rectangles to estimate area. First, you have to
remember the formula for the area of a trapezoid:
With this formula, you can fit trapezoids to the curve in order to estimate the area. For example,
consider the graph of y = 2 + x2, the flight path of Buttercup! Use two trapezoids of equal heights to find
the area under the curve from x=-2 to x=2
Consider the two trapezoids on the graph to the left. The first one
consists of an estimate of the area under the curve from x=-2 to x=0 and
the second one estimates the area from x=0 to x=2. Furthermore,
because these are trapezoids and we know the formula for the area of a
trapezoid, we can calculate this estimated area.
Step 1: Write out the formula for the area of the first trapezoid using
the formula above. The height would be 2, the base1 would be 6, and
the base2 would be 2. Therefore, the area would be:
Step 2: Write out the formula for the area of the second trapezoid. The height would be 2, the base1
would be 2, and the base2 would be 6. Therefore, the area would be:
Step 3: Add up the areas of the two trapezoids to find the estimated area under the curve from x=-2 to
x=2.
Step 4: It is now important to understand your answer. The two trapezoids serve as estimates of the
actual area under the curve of y = 2 + x2. In this case, both trapezoids have an area of 8 (the trapezoids
will not always have equal areas). So, by adding the areas of the trapezoids together, you are obtaining
an estimate of the total area under the curve. In this case, the answer is 16. Now you know the area
under the curve of Buttercup’s flight path!
AP Multiple Choice with…….Johnny Bravo!
Because integrals can be used to find the average value of a function over an
interval, Riemann sums can be used to estimate this average value. The equation
is used to find the average value of a function over an interval is:
that
In
this equation, a represents the starting x value in the integral and b represents the ending x value in the
interval. F(x) is the function in which you are finding the average value. For example, if you were trying
to find the average value of f(x) = x2 + x over the interval [-2,6], your equation would look like:
You would solve this integral just like any other integral, and your answer would give you the average
value of the function over the given integral.
Furthermore, because Riemann sums approximate integrals, Riemann sums can be used to approximate
average values. Using the example above with a left hand Riemann sum of 4 equal sub-intervals,
estimate the average value. Remember, use Johnny Bravo’s massive muscles to inspire you to find the
right answer.
Possible Answers:
a) 14
b)7
c)28
d)17
e)3.5
Step 1: Find the data necessary to complete the left-hand Riemann sum. Because it is a left-hand sum,
you need to know the values of f(x) = x2 + x for x=-2, x=0, x=2, and x=4.
Step 2: Solve the Riemann sum to estimate the integral from x=-2 to x=6. Remember that the sum is
using equal sub-intervals, each of 2. So, to approximate the integral, you must multiply 2 by each of the
values of f(x).
Step 3: Now that you have your approximation, you must apply this to the overall formula for the
average value. You use the 1/(b-a) and multiply this by 56.
Step 4: It is important to understand your answer. By using a Riemann sum, you have essentially
approximated the integral of f(x), and you can then apply this answer to the Average Value formula. The
answer 7 means that on the function f(x) = x2 + x from x=-2 to x=6, the average value of it is 7.
The correct answer is B.
You would get answer A if you multiplied with a height of 4 instead of 2. You would get answer C if you
multiplied with a height of 8 instead of 2. You would get answer D if you used a right hand Riemann
sum. You would get answer E if you forgot to multiply by a height.
Hey Arnold! Since these approximations are in fact approximations, it is worth
discussing whether they are over or under approximations and what this depends
on. Below is a helpful chart to keep organized what conditions produce and over
or under approximation for the different sums. Remember left and right Riemann
sums depend on increasing or decreasing and trapezoidal sums depend on
concavity!
Type of Sum
Condition for over
approximation
Condition for under
approximation
Left-Riemann Sum
f’(x)<0
f’(x)>0
Right-Riemann Sum
f’(x)>0
f’(x)<0
Trapezoidal Sum
f’’(x)>0
f’’(x)<0
*Helga wants you to remember than you cannot easily determine whether a midpoint sum will produce an over or under approximation as it depends on the
specific function or part of the function.
AP Level Conceptual Example
Draw a function that produces an under approximation when a left Riemann Sum is taken, an over
approximation when a right Riemann Sum is taken, and an over approximation when a Trapezoidal Sum
is taken.
Solution:
First identify the characteristics the graph must have to produce the under or over approximations for
specific sums. In this case, there are two that must be met in this case.
1. In order for a left Riemann Sum to be an under and a right Riemann Sum to be an over
approximation the graph must be increasing (f’(x)>0).
2. In order for a Trapezoidal Sum to be an over approximation the graph must be concave up or in
other words the second derivative must be greater than zero (f’’(x)>0).
Example of an Answer:
Left Riemann Sum
Under Approximate
Right Riemann Sum
Over Approximate
Trapezoidal Sum
Over Approximate
Example Utilizing a Graphing Utility!
Cat and Dog from Catdog want me to remind you of a lazy way to
perform Riemann Sums! You are able to utilize your calculator to sum
the heights of the various rectangles. “WAIT!” you say, “You can’t just
add up the heights. You need to multiply the individual heights by the
widths of the rectangles first.” You bring up a good point and
something that must be noted. You can only perform a Riemann sum
on a calculator if the intervals are of equal width. If this is the case then
you can sum the heights through the calculator and multiply by the width that is common to all the
rectangles due to the distributive property!
How awesome and useful of a realization! By using this you can save time performing simple
multiplication and addition that we have all been able to do since elementary school. Here’s an example
below to see how to actually use your calculator to do this useful thing.
Estimate the following integral using a left and right hand Riemann Sum of ten equal subintervals using
your graphing calculator:
To perform the sum on your calculator you will need to use the sum sequence feature. The following
pictures will help you accomplish this:
The Sum button is under the math tab of the stat section of the
calculator.
The seq button is under the ops tab of the state section of the calculator.
Once you have sum(seq( in your calculator you will need to follow it by five things, each separated by a
comma in the following order.
1. The function that you will be integrating/taking the Riemann Sum of
2. The variable or input (almost always will be X) of the function
3. The starting point
4. The ending point
5. The width or length of interval
Number three and four depend on whether a left or right Riemann sum is
being taken and number five varies depending on how long the total
interval is and how many subintervals are requested by the problem. In
this case since the interval is ten long and there are to be ten
subintervals, the fifth will be 1. Now let’s look at how three and four vary
by left or right Riemann sum.
Left Hand Riemann Sum:
The starting point will be x=0 since the left most value of the first interval is zero. The end point will
be one less subinterval unit than the right most point. In this case since the subintervals are one unit
the end point will be x=9. If you think about it, this makes logical sense. The height of the final
interval will be the left most point of that interval.
This is what it will look like in the calculator (remember Y= the function) :
Sum(seq(Y,X,0,9,1)=
Multiply by 1 since this is the width
Right Hand Riemann Sum:
The starting point in this case will be x=1. The first interval is (0,1) and the right most point of this is
x=1. The end point will be x=10 since the final interval is (9,10) and it is a right hand Riemann sum.
This is what it will look like in the calculator:
Sum(seq(Y,X,1,10,1)=
Multiply by 1 since this is the width
Don’t forget in both sums to multiply by the height after the calculator sums the
heights for you!
See how much easier this is than doing all the math out on paper! Cat and Dog’s
laziness
benefit
With Suga Mama from the Proud Family!
Use Suga Mama’s words of wisdom as a
guide on your journey through this epic
problem.
The following table gives values of Suga Mama’s velocity in ft/hr on
her house on her daily morning walk to the IHOP for breakfast. The
looks like it’s going to
you in the long run!
graph gives values of ft/hr in relationship to hours. At t=0, Suga Mama is at position 0 and her velocity is
0 ft/hr.
t (hours)
0
F’(t)
0
(feet/hours)
1
2
3
4
5
6
25
55
100
175
280
400
a) Calculate the following interval using a right hand Riemann
sum of 3 equal sub intervals and explain its meaning using
correct units.
b) Is the answer in part a) an over approximation, under approximation, or neither. Explain.
Explanation:
Part a)
Step 1: Identify the intervals and the values to be used as the heights.
(0,2) F’(2)=55
(2,4) F’(4)= 175 (4,6) F’(6)=400
Step 2: Perform the right hand Riemann sum. Using subintervals of 2, you can estimate the integral of
F’(t).
ft
Step 3: Apply this answer to solve part a. Substitute 2210 for the integral and just multiply by 1/6.
ft/hr
Step 4: Explain the meaning of this answer. Your answer of 210 ft/hr means that the average value of
Suga Mama’s velocity from t=0 to t=6 on her morning walk to IHOP is around 210 ft/hr.
Part b)
Step 1: Explain whether or not the approximation is higher than the actual average value, lower than the
actual average value, or neither. As you learned earlier, whether right and left Riemann sums are over
or under approximations depends on whether the graph is increasing or decreasing. So, you must
determine whether F’(t) is increasing or decreasing on the interval (0,6). To figure this out, look at the
table values of F’(t). Notice that the values of the velocity are constantly increasing. In fact, the rate at
which the velocity is increasing is increasing, so Suga Mama is even accelerating at a positive rate
throughout the interval. (This is because of the increased aroma of pancakes as she moves closer to
IHOP). So, because the function is increasing throughout the interval and a right hand Riemann sum was
used, the estimate for part a is an over approximation.
Remember when taking integrals, derivatives, and average values of distance, velocity or acceleration
functions to be careful with units and pay attention to the way calculus allows you to flow from one type
of function to another. Be sure not to mix them up! The math on a free response such as this will rarely
be too difficult. The difficulty comes in the explanation of the significance of the calculations
(Basic idea and extent of difficult for our AP Free Response question taken from released examples on
College Board)
Famous Mathematician
Bernhard Riemann, a German mathematician, is the man who
invented the Riemann sum. Riemann was born on September 17,
1826 in Hanover, Germany. He studied at the universities of
Gottingen and Berlin, later becoming a professor at Gottingen.
Riemann developed profound and novel approaches to the study of
geometry. He came up with a way to estimate area under a curve
or definite integrals by using rectangles. His methods laid the
mathematical foundation for Albert Einstein’s theory of relativity.
In addition, Riemann made contributions to the theory of
functions, complex analysis, and number theory. There is no doubt
that the man was an extremely important to the history of mathematics. (answers.com)
Connection To The Real World
Integrals, which can be estimated with Riemann and trapezoidal sums, are extremely applicable in the
real-world. Unlike some obscure topics that are learned in upper level math classes, being able to utilize
and complete Riemann sums and trapezoidal sums is a very useful skill to have. First, integrals are not
only able to find areas under a curve on your graphing calculator; they can find the area of virtually
anything. Whether it be the surface area of a lake or the area of a solid of revolution, the possible
applications to today’s society are great. Further, taking integrals can be used to find displacement of
position or the length of a certain path. It can even allow students, who also understand derivatives, to
move back and forth between position, velocity, and acceleration. Certainly it is extremely useful in the
real-world to be able to calculate the velocity of a plane or the length of travel it will take it to reach its
destination. Likewise, there is no denying the usefulness of being able to calculate the difference in
water volume in a sewage plant over time. As you can see, Riemann sums and trapezoidal sums are
useful ways to estimate the integrals that are so important in today’s world.
Works Cited
http://www.answers.com/topic/bernhard-riemann (Bernhard Riemann)
http://mathworld.wolfram.com/RiemannSum.html (Riemann Sum scenarios)
Larson, Hostetler, Edwards. Calculus of a Single Variable. Eight Edition
http://www.math.hmc.edu/calculus/tutorials/riemann_sums/ (Riemann Sums)
College Board. AP Central. AP Calculus.
PRACTICE!
For questions 1-4, use the following table:
x
-2
-1
0
1
2
3
F(x)
3
5
-1
-3
1
2
Estimate the following integrals of F(x) from -2 to 4 using 3 sub-intervals of equal length:
1.
2.
3.
4.
4
6
Using a right hand Riemann sum
Using a left hand Riemann sum
Using a mid-point Riemann sum
Using a trapezoidal sum
For questions 5-8, use the following equation:
5.
6.
7.
8.
Estimate the integral of y from -1 to 5 using a right hand Riemann sum of 3 equal sub-intervals
Estimate the integral of y from 1 to 4 using a left hand Riemann sum of 3 equal sub-intervals
Estimate the integral of y from -3 to 7 using a mid-point Riemann sum of 5 equal sub-intervals
Estimate the integral of y from 3 to 11 using a trapezoidal sum of 4 equal sub-intervals
Estimate the average value of the following functions:
9. f(x) = x2 – 8; from -2 to 4 using a midpoint Riemann sum of 3 equal sub-intervals
10. f(x) = x3 + 2x2 – 3; from -5 to 3 using a trapezoidal sum of 4 equal sub-intervals
Conceptual
11. If a function is increasing and concave up on the interval (0,20) will a left Riemann sum of three
equal sub intervals of 2 units on the interval (0,6) be an over or under approximation? Explain
12. Will a trapezoidal sum taken on the interval (0,10) of the function
be an over or
under approximate? Explain. Does the number or length of intervals matter? Why or why not?
x
1
3
5
7
9
f(x)
1
9
25
49
81
13. Using the above table, can you determine whether a mid point Riemann sum of two equal
subintervals is an over or under approximate? How about a left Riemann of four equal
subintervals? Explain.
Sigma Notation
14. Write the integral that the following sigma notation of a Riemann sum represents. Identify the
kind of Riemann sum and number of sub intervals if possible:
15. For the following integral write the left Riemann sum with five equal length subintervals using
sigma notation:
AP Multiple Choice
1. If
then which of the following are true?
I.
A left Riemann sum is an over approximation.
II.
A right Riemann sum is an over approximation
III.
A trapezoidal Riemann sum is an over approximation
A. I
B. II
C. I and II
D. I and III
E. I, II, III
2. For the graph f= a right Riemann sum produces and over approximation. Which of the
following reasons causes this to be true based on the equation of?
A. f’(x)>0
B. f’(x)<0
C. f’’(x)>0
D. f’’(x)<0
E. f(x) is an even function
x
1
2
3
4
5
6
7
f(x)
8
4
2
1
3
5
10
3. Using the table above approximate the average value on the interval (1,7) using a trapezoidal sum of
six equal sub intervals.
A. 24
B. 48
C. 7
D. 4
E. 8
4. graph of f is represented by
.What is the average value of f on the interval (-2,6)
using a mid-point Riemann sum of four equal subintervals?
A. 752
B. 188
C. 1504
5. The expression
which of the following?
A.
D. 220
E. 94
is a Riemann Sum approximation for
B.
C.
D.
E.
Which of the following is the correct sigma notation for a right Riemann sum of six equal subintervals of
2
the following interval:
 xdx
3
0
9
A.

n 0
3
n
5
10
B.

n 0
3
n
5
C.
1 9 3 n

5 n 0 5
D.
1 10 3 n

5 n1 5
9
E.
 n
3
n 0
AP Free Response
Otto and the Rocket Power gang are surfing out in the ocean. A huge wave begins to form in
the distance. The following table gives values for the height of the waves in relationship to its
width from the left.
Distance from left
of wave (feet)
0
7
18
24
36
44
53
Height of wave
(feet)
0
5
13
26
16
7
0
a) Use a trapezoidal sum with the six sub-intervals indicated by the data in the
table to approximate the area of the face of the wave. Show correct units.
b) Use your answer from part a to estimate the average value of the height of the
wave from 0 feet from the left to 53 feet from the left. Is this value an under
approximation, over approximation, or is it indeterminable?
c) The function
gives values of the wave’s velocity as it moves
toward shore. If at time t=11 the wave is 0 feet from the shore, and at time t=0
the wave is 50 feet from shore, how far away from the shore is the wave at t=5?
d) At what time is the wave farthest from the shore? Why does this make sense in
terms of how a wave moves in relationship to the shore?