Case Study: Oil Prices
Resource Material: Leahy, M. (2014, October 8). U.S. Energy Information
Administration - EIA - Independent Statistics and Analysis. Retrieved May 18,
2015, from http://www.eia.gov/todayinenergy/detail.cfm?id=18311
Learning Goals: The learning goals of this case study include using both Riemann
Sums and Trapezoidal Sums to approximate accumulated change from a rate of
change and interpret the answer in terms of the contextual situation.
Warm-Up Exercises for Calculus Case Study
1. With a Right-Hand Riemann Sum, approximate the area between the graph of
𝑥2
the function 𝑓(𝑥) = 5 and the x-axis on the interval [0,10] using 10 subintervals of equal width.
2. With a Left-Hand Riemann Sum, approximate the area between the graph of the
function 𝑓(𝑥) =
equal width.
𝑥2
5
and the x-axis on the interval [0,10] using 10 sub-intervals of
1
3.
With a Trapezoidal Sum, approximate the area between the graph of the
function 𝑓(𝑥) =
equal width.
𝑥2
5
and the x-axis on the interval [0,10] using 10 sub-intervals of
2
3
Study Questions for Calculus Case Study
1.
Using information provided in the graph from the U.S. Energy Information
Administration, approximate the total number of crude oil barrels produced
in Libya during each of the following:
a.
June, 2014
b.
July, 2014
c.
August, 2014
2.
The graph from the U.S. Energy Information Administration also provides
data regarding the dollars in revenue per barrel. Using the graph,
approximate Libya’s revenue during each of the following:
a.
June, 2014
b.
July, 2014
c.
August, 2014
3.
How do your approximations differ from your classmates? What was
different in your approaches to obtaining the approximations?
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Warm-Up Exercises with Answers
1. With a Right-Hand Riemann Sum, approximate the area between the graph of
𝑥2
the function 𝑓(𝑥) = 5 and the x-axis on the interval [0,10] using 10 subintervals of equal width.
RHR Sum =
1
4
9
16
36
49
64
81
1( ) + 1( ) + 1( ) + 1( ) + 1(5) + 1( ) + 1( ) + 1( ) + 1( ) + 20
5
5
5
5
5
5
5
5
= 77
2. With a Left-Hand Riemann Sum, approximate the area between the graph of the
function 𝑓(𝑥) =
equal width.
𝑥2
5
and the x-axis on the interval [0,10] using 10 sub-intervals of
LHR Sum =
1
4
9
16
36
49
64
81
1(0) + 1( ) + 1( ) + 1( ) + 1( ) + 1(5) + 1( ) + 1( ) + 1( ) + 1( )
5
5
5
5
5
5
5
5
= 57
3.
With a Trapezoidal Sum, approximate the area between the graph of the
function 𝑓(𝑥) =
equal width.
𝑥2
5
and the x-axis on the interval [0,10] using 10 sub-intervals of
5
Study Questions with Sample Answers
NOTE: Draw students’ attention to the use of proper units when discussing the
numbers.
To obtain units of “barrels” in Question 1, students should show the height of the
1000 𝑏𝑎𝑟𝑟𝑒𝑙𝑠
rectangles have a unit of
, while the width of the rectangle can have units
𝑑𝑎𝑦
of 𝑑𝑎𝑦𝑠. Thus, we have
1000 𝑏𝑎𝑟𝑟𝑒𝑙𝑠
𝑑𝑎𝑦
∗ 𝑑𝑎𝑦𝑠 = 1000 𝑏𝑎𝑟𝑟𝑒𝑙𝑠.
To obtain units of “dollars” in Question 2, students should understand
1000 𝑏𝑎𝑟𝑟𝑒𝑙𝑠
𝑑𝑜𝑙𝑙𝑎𝑟𝑠
∗ 𝑑𝑎𝑦𝑠 ∗
= 1000 𝑑𝑜𝑙𝑙𝑎𝑟𝑠
𝑑𝑎𝑦
𝑏𝑎𝑟𝑟𝑒𝑙
1. Using information provided in the graph from the U.S. Energy Information
Administration, approximate the total number of crude oil barrels produced
in Libya during each of the following:
a. June, 2014
Using a Right-Hand Riemann Sum with one interval of width 30 days,
1000 𝑏𝑎𝑟𝑟𝑒𝑙𝑠
RHR SUM = 300
∗ 30 𝑑𝑎𝑦𝑠 = 9,000,000 𝑏𝑎𝑟𝑟𝑒𝑙𝑠
𝑑𝑎𝑦
Thus, an approximation for the number of crude oil barrels produced in Libya in
June, 2014, is 9,000,000 barrels. This is an over-estimate.
b. July, 2014
Using a Right-Hand Riemann Sum with one interval of width 31 days,
1000 𝑏𝑎𝑟𝑟𝑒𝑙𝑠
RHR SUM = 450
∗ 31 𝑑𝑎𝑦𝑠 = 13,950,000 𝑏𝑎𝑟𝑟𝑒𝑙𝑠
𝑑𝑎𝑦
Thus, an approximation for the number of crude oil barrels produced in Libya in
July, 2014, is 13,950,000 barrels.
c. August, 2014
Using a Right-Hand Riemann Sum with one interval of width 31 days,
1000 𝑏𝑎𝑟𝑟𝑒𝑙𝑠
RHR SUM = 700
∗ 31 𝑑𝑎𝑦𝑠 = 21,700,000 𝑏𝑎𝑟𝑟𝑒𝑙𝑠
𝑑𝑎𝑦
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Thus, an approximation for the number of crude oil barrels produced in Libya in
August, 2014, is 21,700,000 barrels. This is an over-estimate.
2.
The graph from the U.S. Energy Information Administration also
provides data regarding the dollars in revenue per barrel. Using the graph,
approximate Libya’s revenue during each of the following:
a.
June, 2014
𝑑𝑜𝑙𝑙𝑎𝑟𝑠
9,000,000 𝑏𝑎𝑟𝑟𝑒𝑙𝑠 ∗ 110 𝑏𝑎𝑟𝑟𝑒𝑙 = 990,000,000 𝑑𝑜𝑙𝑙𝑎𝑟𝑠
b.
July, 2014
13,950,000 𝑏𝑎𝑟𝑟𝑒𝑙𝑠 ∗ 105
1,464,750,000 𝑑𝑜𝑙𝑙𝑎𝑟𝑠
August, 2014
21,700,000 𝑏𝑎𝑟𝑟𝑒𝑙𝑠 ∗ 100
2,170,000,000 𝑑𝑜𝑙𝑙𝑎𝑟𝑠
𝑑𝑜𝑙𝑙𝑎𝑟𝑠
𝑏𝑎𝑟𝑟𝑒𝑙
=
c.
𝑑𝑜𝑙𝑙𝑎𝑟𝑠
𝑏𝑎𝑟𝑟𝑒𝑙
=
3.
How do your approximations differ from your classmates? What was
different in your approaches to obtaining the approximations?
Answers may vary depending on whether students used a Left-Hand Riemann Sum,
Right-Hand Riemann Sum, or a Trapezoidal Sum.
Classroom discussion should include comparing students’ methods and identifying
whether the approximations appear to be over-estimates or under-estimates, as
well as a discussion of who most likely obtained the most accurate approximation
(and why, ie. number of rectangles/trapezoids used, least amount of white space
between the curve and the rectangles/trapezoids).
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