This project helps students with the concept of using the rectangle method to determine the area under a function:

Goal:

Work in Groups to find a solution for finding the area under a curve.

Group 1 - use rectangles as “pieces,” with the left side of each interval to determine the height

Group 2 - use rectangles as “pieces,” with the right side of each interval to determine the height

Group 3 - use rectangles as “pieces,” with the midpoint side of each interval to determine the height

Group 4 - use trapezoids as “pieces” (rectangles + triangles)

Course Outcomes addressed:

• Understand integrals visually as area under a function (definite integral) and as a process of reversing derivatives (antiderivative).

• Calculate integrals of elementary functions and compositions of elementary functions.

• Recognize integrals in the practical and professional world around them, particularly in environmental and life science.

• Work in groups … and communicate the results of mathematical investigations ...

Suggested Process/Methods:

1. First use the curve f(x) = x^2 from 0 to 1 for #2-6 below, then consider the same 2-5 for f(x) = exp(x^2) from 0 to 1.

2. Find the exact area under the curve using WolframAlpha.com (you can query in natural language).

3. Draw a picture of the curve, and break it into 4 pieces.  Find the formula that yields an approximation to the area under the curve.  Compute and compare this to the value that you achieved in part 1.  What is the error?

4. Repeat for 8 pieces and compare the value to the true value.  What is the error?

5. Write a formula that breaks up the area into n pieces.  What is the width of the each piece when you use n pieces?

6. Now consider the limit as n -> infinity.  What is the limit of the width of each piece as n-> infinity?  Can you compute the limit of the area equation as n-> infinity?

7. Finally consider any function, f(x), from x=a to x=b broken into n intervals.  Simplify as much as possible.  Write the formula that best approximates the area under this curve.

Communicating the results:

Group presentation to class on in one week, showing your peers the answers to the above questions and the process to get there.  The presentation may be a board presentation, but you are expected to provide a handout.  Alternatively, you may present a short powerpoint.  Each group should aim for a presentation in the neighborhood of 7 minutes, with each member contributing equally to each part of the presentation.  The grading rubric is on Canvas.