Exploring approximations of definite integrals


In this applet you can enter your own function, left bound, and right bound using the appropriate input boxes. Then, you can click the left hand sum, right hand sum, and trapezoid rule to explore these approximations of the definite integral for your function on your bounds. You can also change the number of intervals by adjusting the slider n (i.e., n = the number of rectangles or trapezoids used to make the approximation). You can also see the exact value of the definite integral on your bound by clicking "Show Integral". Play with different functions, different bounds, and/or different numbers of intervals and make some observations. After you have had some time to explore, consider the questions below the applet.  
Some questions you can consider as you play with different functions and bounds: How are each of the sums calculated (e.g., how are the rectangles or trapezoids drawn)? What value is used for the height(s) of the shape? What value is used for the width of the shape? What happens as n (the number of intervals) gets larger for each case? What size n would be enough to get a good approximate for each of your functions and bounds for each of the 3 sums? How do the left sum and right sum compare on functions that are always increasing? Always decreasing? Does your statements hold if the function is always increasing/decreasing? How does the sum from the trapezoid rule compare to the left sum and right sum? Is there any relationship you can generalize about these three sums?