As a high school or undergraduate Mathematics teacher, you can use this set of computerbased tools to help you in teaching topics such as integration, definite integral, area under a curve, and Riemann sum in Introductory Calculus.
This lesson plan allows students to understand Riemann sum, calculate the area under a curve using Riemann sum, and explore how this value converges to a definite integral. The activity helps students to apply the Riemann sums method for analysis and comparison of data on CO_{2} emission, which is considered to be a significant contributor to climate change.
Thus, the use of this lesson plan allows you to integrate the teaching of a climate science topic with a core topic in Mathematics.
Questions
Use this lesson plan to help your students find answers to:
 1. For a given function f(x) and n, calculate the left Riemann sum and right Riemann sum.
 2. For the same f(x) (as above) and double the value of n (from above), calculate and compare the left and right Riemann sums.
 3. Using Riemann sums, calculate and compare the total CO2 emissions (data records provided in the activity) for the U.S. and China from 1980 to 2015. What are the possible effects of these CO2 emissions on the Earth’s climate?
About Lesson Plan
Grade Level  High School, Undergraduate 
Discipline  Mathematics 
Topic(s) in Discipline  • Calculus, Integration • Definite Integral • Area under a Curve • Riemann Sum 
Climate Topic  • Energy, Economics, and Climate Change • Climate and the Anthroposphere • Policies, Politics, and Environmental Governance 
Location  USA and China 
Languages  English 
Access  Online, Offline 
Approximate Time Required  90 – 120 min 
Contents
Reading and Associated Activity
(30 – 60 min) 
A reading that introduces Riemann sum and the types of Riemann sums. It describes the calculation of the area under a curve by using Riemann sum, and explains how this value can converge to the definite integral. https://mathinsight.org/calculating_area_under_curve_riemann_sums 
Classroom/ Laboratory activity
(~60 min) 
A classroom/laboratory activity to analyze CO_{2} emissions data for the U.S. and China by using Riemann sums for the calculation of area under the curve. http://sustainabilitymath.org/calculusmaterials/ 
Here is a stepbystep guide to using this lesson plan in the classroom/laboratory. We have suggested these steps as a possible plan of action. You may customize the lesson plan according to your preferences and requirements.
1. Introduce the topic by using a reading and an associated activity 

2. Conduct Classroom/ Laboratory activity (~60 min) 

Use the tools and the concepts learned so far to discuss and determine answers to the following questions:
 1. For a given function f(x) and n, calculate the left Riemann sum and right Riemann sum.
 2. For the same f(x) (as above) and double the value of n (from above), calculate and compare the left and right Riemann sums.
 3. Using Riemann sums, calculate and compare the total CO2 emissions (data records provided in the activity) for the U.S. and China from 1980 to 2015. What are the possible effects of these CO2 emissions on the Earth’s climate?
The tools in this lesson plan will enable students to:
 • calculate the approximate area under a curve by using the Riemann sums method
 • compare the results obtained for left and right Riemann sums for the same curve
 • explain how the estimate of the area under a curve (using Riemann sum) converges to the definite integral
 • apply the Riemann sum method to analyze and compare CO_{2} emissions data for the U.S. and China
Videos  Video tutorial, “Riemann sums”, from Khan Academy: https://www.khanacademy.org/math/oldintegralcalculus/riemannsumsic 
1  Reading and Associated Activity, “Calculating the area under a curve using Riemann sums”  D. Q. Nykamp from Math Insight 
2  Classroom/Laboratory Activity, “U.S. and China CO2 Emissions”  Thomas J. Pfaff , Sustainability Math 
3  Additional Resources  Khan Academy 
4.  Images  https://matplotlib.org/gallery/showcase/integral.html https://math.stackexchange.com/questions/1180034/inequalityandintegral 
Integral as the area under a curve
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