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