**Prerequisites:** Limit, Derivative

**Goal:** To understand Riemann Sums

- Given a curve drawn on graph paper, what is the fastest way to estimate the area under the curve?
- How can you get a better estimate of the area?
- Other than squares, what is another easy shape to find the area of?
- How can you use rectangles to estimate the area under a curve?
- How can you make that estimate better and better?
- Using the controls on the right in the applet above, chose the function
*y=2.0*and the "right endpoint" rule. Using the sliders below the graph, choose left and right endpoints^{x}*-2.0*and*1.0*. Using the slider on the left, estimate the area under the curve using 2 rectangles. - What is the estimated area? (It should appear in black at the bottom of the graph.)
- Will your answer in part
*a*be an under- or over-estimate of the actual area? Why? - What is the width of each rectangle?
- What are the
*x*-coordinates of the edges of the rectangles? - What are the heights of the rectangles? (Hint: you will need
*2*of the*3*values in step d.) - What are the areas of the rectangles?
- 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.
- Repeat step 6 using
*5*rectangles. It might help to use summation notation. - Repeat step 6 using
*n*rectangles. You will need summation notation, and skip part a. - How can we transform the answer to question 8 into the
*exact*area under the curve instead of a mere estimate? - Repeat step 6 using the "left endpoint" rule and
*5*rectangles. - Repeat step 6 using the "left endpoint" rule and
*n*rectangles. Skip part a. - How can we transform the answer to question 11 into the
*exact*area under the curve instead of a mere estimate? - Leibnitz used a long S, written ∫ , to indicate the limit of the summation. This is our symbol for what?