I started by tracing my hand onto the graph paper, and then marked the data points using midpoint Reimann's sums. I entered the data points into my calculator and found the area of the function. To account for the fact that the outline was not a function, I divided the outline into three separate lines that were each individual functions; I had an overall function and two functions for areas underneath my hand. I calculated the Reimann's sums for each function, then subtracted the areas underneath from the overall to find the actual area of my hand (see diagram and work).

The area of the overall function was 19.75975 square inches; the areas of the two functions underneath were 1.111 square inches and 0.1595 square inches. The area of my hand is 18.48925 square inches. I used the general equation R = *x [ f(a) + f(a+*x) + f(a+2*x) + ... + f(b)], where *x equals the horizontal interval between data points taken, for the function f(x) in the interval from x=a to x=b.

Due to the large number of intervals and irregularity of the shape it is very hard to tell whether I over- or underestimated the area. The Reimann's intervals follow the outline very closely, but the area does seem to be very slightly overestimated.

I used very fine-grid graph paper (.05 inch intervals) to increase my accuracy. I also "corrected" slightly the outline of my hand, because there were often nearly-vertical places in the outline that did not line up with the grid and would lead to inaccurate midpoints. If the outline did contain a vertical line along the grid, I did not use that part of the outline in finding a midpoint; I only used the sections of outline between the grid lines. This also led to more accurate midpoints.

Possible sources of error include inaccuracies in tracing and guessing the intersection of the outline with the grid lines, making it hard to determine the interval for which the midpoint must be found. Other sources of error include misreading the very small areas of the graph and the possibility of copying the long strings of numbers onto paper and into the calculator incorrectly.

If I was to do this project again I would try to find a way to draw the outline more precisely. Although I traced and marked the midpoints with a thin mechanical pencil, it was often hard to determine where the blurred outline intersected the grid lines on the paper. The slope of the line was often very steep or almost vertical, especially along the sides of my fingers, making it even harder to find exactly where the outline intersected the grid. This made it hard to find the midpoint. I did not have any problems with making calculations or any other part of the project.