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 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
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.