# Computing upper and lower Riemann sums

Compute the lower and upper Riemann sums L(f, P$$_n$$) and U(f, P$$_n$$) for the following functions f(x) and the partition P$$_n$$ = {0, $$\frac{1}{n}$$, ..., $$\frac{n}{n}$$ = 1} of [0, 1]

I find the Riemann sums in general to just be very confusing

a) f(x) = x + 3

b) f(x) = 3 - x

c) f(x) = e$$^x$$

• what have you tried? Jun 14 '20 at 6:18

## 1 Answer

You are given three functions and a sequence of points and are asked merely to evaluate the functions at those given points. All of the functions in question are continuous, so this is relatively straightforward.

For a) we have $$L(f,P_n) = \sum_{i=0}^n \frac in \cdot3\cdot \frac in = n+\frac{1}{2 n}-\frac{3}{2}, \ \ U(f,P_n) = \sum_{i=0}^n \frac in \cdot 3\cdot\frac {i+1}n = n-1/n,$$ and the other functions can be handled in the same manner.

However, I question the pedagogical value of emphasizing the explicit evaluation of Riemann sums, when there are more pressing issues such as determining necessary and sufficient conditions under which the integral exists, and deriving such fundamental theorems as...well, the fundamental theorem of calculus.