Let f (x) = x^{2} on the interval [0,12]. How many rectangles must we use to guarantee that the right- and left-hand sums are within 100 of each other?

Having the right- and left- hand sums within 100 of each other means we want

From the problem we can tell that we want to have a = 0 and b = 12, so

f (a) = f (0) = 0

and

f (b) = f (12) = 144.

Plugging all these numbers into the inequality, we want

We solve this inequality for n:

We conclude that we need to use at least 18 rectangles, since we need a whole number of rectangles.