Lesson Plan: Teaching Introductory Calculus (Integration) by using CO2 Emissions Data

Here is a step-by-step 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

• Introduce the topic of Riemann sum and the types of Riemann sums by using the reading, “Calculating the area under a curve using Riemann sums” from D. Q. Nykamp at Math Insight. • Conduct the activities using the applets to further explain how to calculate the area under a curve by using the Riemann sum and how this value can converge to the definite integral. •  The reading “Calculating the area under a curve using Riemann sums” and the associated activities (using applets) are available at https://mathinsight.org/calculating_area_under_curve_riemann_sums

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

• Then, help your students apply the learned concepts through a hands-on classroom/laboratory activity, “U.S. and China CO2 Emissions”, by Thomas J. Pfaff at Sustainability Math. This activity uses CO2 emission data and population data for the U.S. and China for the period 1980 to 2015. • This activity will help students to • analyze CO2 emissions for each country •  compare the CO2 emissions for the countries by using Riemann sums for the data from 1980 to 2015 •  create a proposal for emission reduction by considering past and current CO2 emissions for the two countries •  Go to http://sustainabilitymath.org/calculus-materials/ • Download the material in the project, “U.S. and China CO2 Emissions”, under Calculus I – Integration Related Projects. •  Students can perform the exercises described in the Word file by using the data in the Excel file.

 

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 CO₂ emissions data for the U.S. and China

If you or your students would like to explore the topic further, these additional resources will be useful.

Videos

Video tutorial, “Riemann sums”, from Khan Academy: https://www.khanacademy.org/math/old-integral-calculus/riemann-sums-ic

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/inequality-and-integral

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