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Riemman Sum

Rectangle Approximation for Area (Reimann Sum)

The velocity (ft/sec) of a ball is graphed to the right. The area between the curve, the x-axis, and the lower & upper limit can be estimated using rectangles of equal width as shown.

1. Adjust the number of rectangles & look at the "shaded area" numbers. Do more rectangles produce a better estimate for the area between the curve and the x-axis. How do you know?

2. What is the formula for the width of each rectangle in terms of the lower limit, upper limit, & the number of rectangles?

3. What is the height of each of the rectangles?

4. Adjust the number of rectangles to 15. Would you agree that the velocity is "almost" constant over each time interval for each rectangle? Why would the formulas A=lw & d = rt really mean the same thing for each of these rectangles if the velocity was constant over each time interval?

5. Use the "trace" dot to find the starting height, ending heigth, and change in position for the ball. What do you think the shaded area would be if you used 1000 rectangles (hint: think about question 4)?

6. Does the velocity curve indicate that the ball will go up, down, or both ("reset time" & "throw ball" to check your answer)?

7. Sometimes it is convenient to think of a continuous velocity curve as "almost" a bunch of small time intervals of constant velocity. Set the number of rectangles to 3, click "piecewise velocity", "reset time", & "throw ball". Now click "continuous velocity", "reset time", & "throw ball". Is there much difference in the motion? Is there a noticable difference when you use 10 rectangles?

8. Can you get the "shaded area" value to be negative. What does this mean in terms of the ball? Throw the ball to check.