Problem

Calculate

using the limits of Riemann sums.

Solution

Let P_i be the partition of [0, 1] constructed by subdividing [0, 1] in half i times. The first three partitions look like this:

P₀ =

P₁ =

P₂ =

The subintervals of partition P_i have length

so

and in the general case,

This equals

We use the fact from Example 3 of section 4.2 that the sum of the first n positive integers is

(here with n = 2^N – 1).

According to Theorem 1, section 4.2,

where c_i is any point in the ith subinterval, the number of subintervals gets large and the length of the largest subinterval approaches zero.

Here, for L(P_N, f), if we let N approach infinity then the number of subintervals, n, will also approach infinity and ||P||, the maximum length of the subintervals, will approach zero. So, assuming f(x) = x is integrable,

Since f(x) = x is a polynomial function, it is continuous and thus integrable. So the integral is indeed equal to one half.

Checking the answer

There are 2 ways to check our answer. The first is to calculate

and see if we get the same value. Here, we'll evaluate f(x) = x at the rightmost point of each subinterval to get the maximum value.

This equals

As N approaches infinity,

The second way to check is to look at the graph of f(x) = x.

This is a triangle with base = 1 and height = 1, so its area is