The student conjectured that every time , or the number of rectangles in the interval, doubles, the difference between the upper sum and lower sum halves. After exploring the applet below by experimenting with different functions and number of rectangles, the student's conjecture seems to only be true when the function is monotonic on the given interval. If a function is monotonic and decreasing, then the height of the lower sum of the rectangle is given by . However, this quantity will also represent the height of the rectangle in the upper sum. Furthermore, the width of each rectangle is given by . Thus the area of each rectangle is given by . Since the height of each rectangle within is repeated in both the upper and lower sums, the difference between the areas of these rectangles will be zero. Thus, the only terms left after the subtraction of the upper and lower sums will be . By factoring we get that the difference between the upper and lower sums is . Now suppose that the number of intervals is doubled. Then the width of each rectangle is given by and the difference between the upper and lower sums will be . Thus, when the number of rectangles is doubled, the area is halved.
If the function is monotonic and increasing, then a similar argument will hold.
If the function is not monotonic, such as , then the argument will not hold because it is not always true that will represent the upper and lower sums of adjacent rectangles.