Activity Overview
In this Derive™ activity, students investigate the Fundamental Theorem of Calculus and explore examples of Riemann Sums for approximating the Definite Integral: the Midpoint Sum, the Left Hand Endpoint Sum, the Right Hand Endpoint Sum, The Trapezoidal Sum, and Simpson's Approximating Sum.
Before the Activity
See the attached DFW file for detailed instructions for this activity
Print pages from the attached DFW file for your class
During the Activity
Distribute the pages to the class.
Follow the Activity Procedures:
Set up a 2D plot window
Construct the expressions for the Midpoint Sum and other Riemann Sums, in stages
Plot a polynomial function
Observe an expression whose graph is a sequence of rectangles over an interval having the value of the function at the midpoint of each interval as the height of the rectangle above that interval
Graph the result
Evaluate the area lying between the resulting graph and the x-axis
Realize that if a function is positive over an interval, the Midpoint Sum is the area of a sequence of rectangles (constructed as illustrated in the graph)
Notice that the area of the rectangles is almost the same as the area between the graph and the x-axis
Understand that the definite integral of a function over an interval is the difference of the area of the region lying above the x-axis and below the graph of the function, and the area of the region lying below the x-axis and above the graph of the function
Use the Derive software to compute the value of an integral over an interval, and compare the result with the value calculated with the help of the Midpoint Sum formula
Graph a polynomial function
Set the lower value of the interval as a constant and let the upper value be a variable
Use the formula for the Midpoint Sum to approximate the integral
Record the action of the Midpoint Sum when the value of the function is zero, negative, positive, minimum, and maximum
Understand that as the limit of the maximum width of the subintervals making up the partition goes to zero, the variable sum will have the function as its derivative
After the Activity
Students answer questions listed on the activity sheet.
Review student results:
As a class, discuss questions that appeared to be more challenging
Re-teach concepts as necessary