**Prerequisites:** Derivatives, Summation Notation.

**Goal:** To gain familiarity with the concept of using a Riemann
Sum to estimate the area under a curve. To conjecture the area under the
curve *f*(*x*)=*x*^{3} from 0 to 2.

Today we begin by starting Maple, then copying and pasting the following code into our Maple worksheet. The color is only for your own reference; you may want to edit the purple characters, but probably not the red.

restart: with(plots): with(student): xMin := 0: xMax := 2: xSub := 5: # intervals f := x -> x^3: # function leftbox(f(x), x=xMin..xMax, xSub); evalf(leftsum(f(x), x=xMin..xMax, xSub));After pasting this code into Maple, press <enter> (or <return>). The graph of the function should appear, along with 5 rectangles whose heights are obtained by using the height of the left hand endpoint of each sub-interval. Below the graph, the sum of areas of rectangles appears.

- How is the sum related to the area under the curve? I.e., is it greater than, less than, or equal to the area under the curve (between 0 and 2)? Why?
- What is the width of each rectangle? What are the heights of the 5 rectangles? (Hint: the first one is a
*degenerate*rectangle.) - Change the number of sub-intervals (xSub) to 13. How does this change the sum? Will the new sum be closer to or further from the area under the curve? Why?
- Repeat step 3 with the number of sub-intervals equal to 200. How does this change the sum? Will the new sum be closer to or further from the area under the curve? Why?
- Change xSub back to 5, the word "leftbox" to "rightbox," and the word "leftsum" to "rightsum." The new graph should use the right hand endpoint of each sub-interval to obtain the height of each rectangle. How will this change the sum?
- Repeat steps 2-4.
- Make a conjecture about the area under the curve.
- Repeat steps 1-7 for the function
*f*(*x*)=sin(*x*) on the interval 0≤*x*≤π.