All Boxed Up

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)=x3 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.
  1. 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?
  2. What is the width of each rectangle? What are the heights of the 5 rectangles? (Hint: the first one is a degenerate rectangle.)
  3. 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?
  4. 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?
  5. 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?
  6. Repeat steps 2-4.
  7. Make a conjecture about the area under the curve.
  8. Repeat steps 1-7 for the function f(x)=sin(x) on the interval 0≤x≤π.
Last Update: 2007/06/25