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.

Example 2

Let f(x) = x^{2} on the interval [0, 12]. How large must n be to guarantee that RHS(n) is within 0.5 of the exact area between f and the x-axis on [0, 12]?

This question is asking the same sort of thing as the previous one. Since f is increasing on this interval, the exact area must be between LHS(n) and RHS(n). If we can get LHS(n) and RHS(n) within 0.5 of each other, then RHS(n) and LHS(n) will each be within 0.5 of the exact area.