Problem
Calculate
using the limits of Riemann sums.
Solution
Let Pi 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 Pi 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 = 2N – 1).
According to Theorem 1, section 4.2,
where ci 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(PN, 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