Prerequisites: Limit, Derivative

Goal: To understand Riemann Sums

The applet above shows a graph and allows the user to estimate the (signed) area "under" the graph using four different methods. There are many different estimation methods, and only four of the most common ones are implemented in the applet. We will only use two of them.
  1. Given a curve drawn on graph paper, what is the fastest way to estimate the area under the curve?
  2. How can you get a better estimate of the area?
  3. Other than squares, what is another easy shape to find the area of?
  4. How can you use rectangles to estimate the area under a curve?
  5. How can you make that estimate better and better?
  6. Using the controls on the right in the applet above, chose the function y=2.0x and the "right endpoint" rule. Using the sliders below the graph, choose left and right endpoints -2.0 and 1.0. Using the slider on the left, estimate the area under the curve using 2 rectangles.
    1. What is the estimated area? (It should appear in black at the bottom of the graph.)
    2. Will your answer in part a be an under- or over-estimate of the actual area? Why?
    3. What is the width of each rectangle?
    4. What are the x-coordinates of the edges of the rectangles?
    5. What are the heights of the rectangles? (Hint: you will need 2 of the 3 values in step d.)
    6. What are the areas of the rectangles?
    7. Write, as a sum (in other words, leave the "+" in the expression), the total area of the rectangles; note that this is an estimate of the area under the curve. Check that your sum matches your answer in part a.
  7. Repeat step 6 using 5 rectangles. It might help to use summation notation.
  8. Repeat step 6 using n rectangles. You will need summation notation, and skip part a.
  9. How can we transform the answer to question 8 into the exact area under the curve instead of a mere estimate?
  10. Repeat step 6 using the "left endpoint" rule and 5 rectangles.
  11. Repeat step 6 using the "left endpoint" rule and n rectangles. Skip part a.
  12. How can we transform the answer to question 11 into the exact area under the curve instead of a mere estimate?
  13. Leibnitz used a long S, written ∫ , to indicate the limit of the summation. This is our symbol for what?
Last update: 2007/07/30