Integral Calculus, Upper and Lower Sums

Upper and Lower Sums

Place a net of granularity δ across an interval, or throughout a rectangular volume if you prefer. Compute the Riemann sum as before, but this time use the minimum value of f on each subinterval, instead of using f(x_i). This is called a lower sum, because it is always less than or equal to the Riemann sum. If f does not attain its minimum, use the lower bound of f on each subinterval. If f is unbounded (below), the lower sum is not defined.

To compute the upper sum, use the maximum value of f on each subinterval. This is greater than or equal to the Riemann sum.

In the earlier example involving the parabola, our Riemann sums were actually lower sums. This is because x² is increasing on the interval [0,1], hence f(x_i) is always the minimum value of f over its subinterval.

If you add a point to a preexisting net, it cuts one of the subintervals into two pieces. This might increase the minimum value on one of the two pieces. Thus the lower sum can only increase. Similarly, the upper sum can only decrease. Use induction to show that the union of two nets can only increase the lower sum of each net, and decrease the upper sum of each net.

Show that the above results apply in higher dimensions. For any net, lower sum ≤ Riemann sum ≤ upper sum.