Upper and Lower Sums of Equations

Lane Tait
This applet allows you to determine the area underneath all functions. Feel free to type in any function your heart desires into the Input box and then analyze the difference between the Upper and Lower sums of the function you made. There are questions below to help aid your understanding.
1) After typing in your function, adjust the "low" and "up" sliders however you want. 2) Check the box that says "Show Lower Sum" and adjust the "nrect" slider to change the amount of rectangles that are used to compute the sum. 3) Check the box that says "Show Area Under f". What do you notice about the area under the curve and the Lower Sum when you increase the number of rectangles? 4) Now uncheck the "Show Lower Sum" and check the "Show Upper Sum" box. What happens to the Upper Sum when you increase the number of rectangles? 5) Do the Upper and Lower Sums converge to a particular value? 6) If we had the ability to produce and infinite amount of rectangles, what do you think the values Upper and Lower Sums will be? Will they be equal?