Area Under the Curve
The following applet allows you to find the area under the curve using an upper sum, a lower sum, or finding the actual area under the curve for any function. (ex. sin(x)) You must set the limits of integration, a and b, for the applet to find the area over the given interval. You also may set the number of rectangles for the upper/lower sums aproximation.
Consider the following questions:
1) What happens to the value of the upper and lower sums as you increase the number,n, of rectangles? What value does it approach?
2) Choose a value of n, then find the upper and lower sum over an interval of your choosing. What is the average between the upper and lower sum? What value is this close too?
3) For the function sin(x), what are the values of the upper and lower sums over the interval 0 to 2pi? How do these relate to the actual area under the curve?