Let f (x) = x2 on the interval [0,12]. How many rectangles must we use to guarantee that the right- and left-hand sums are within 10
of each other?
Since a, b, f (a), and f (b) are the same as in the example, we get the inequality
We solve this inequality for n:
We conclude that we need to use at least 173 rectangles.
Remember that if LHS(n) and RHS(n) are close to each other, they must also be close to the exact area between f and the x-axis on [a,b].