 As a high school or undergraduate Mathematics teacher, you can use this set of computer-based 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 CO2 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

• 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?

### Left-handed and Right-handed Riemann sums of an increasing function 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 CO2 emissions data for the U.S. and China by using Riemann sums for the calculation of area under the curve. http://sustainabilitymath.org/calculus-materials/

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.

### Integral as the area under a curve 