# Find the Upper and lower sums on a regular partition of the intervals, for the following integral:

Original question

$$\int_1^2 f(x)\,\mathrm{d}x$$ where $$f(x)=\begin{cases} 1 & \text{if x is rational,} \\ 0 & \text{if x is irrational} \end{cases}$$

How does one interpret this to find the upper and lower sums on a regular partition

$$\Delta x={1\over n}$$

• The values of x are 0, 1/n, 2/n, 3/n, ..., (n-1)/n, 1. Each pair of successive x values, 0 to 1/n, 1/n to 2/n, etc. gives an interval. To find the "upper sum" take the largest value of f in each interval. To find the "lower sum", take the lowest value of f in each interval. – user247327 Apr 18 '16 at 12:39