Riemann sums and partitions
This applet shows the lower sum $L(f,P)$ and upper sum $U(f,P)$ for a function $f$ and partition $P$.
Drag the points A and B on the x-axis to change the endpoints of the partition. Click ‘Add another point to partition’ to refine the partition. The new point in the partition appears in yellow. Drag it to move it. Repeat this to further refine your partition. What happens to the lower and upper sums as you refine the partition?
The lower pane shows a plot of the lower and upper sums vs the partition size $n$. What happens to the lower and upper sums as $n$ increases? Is the function Riemann integrable?
Some interesting functions to try:
- A polynomial, eg. $f(x) = x(x-2)(x-3)$
- $f(x) = 2$ (a constant function).
- $f(x) = \sin(x)$. Try between 0 and 2π.
- A monotonic bounded function $f(x) = \lfloor x \rfloor$.
Enter this as
f(x) = floor(x) - The Dirichlet function
$f(x) = \begin{cases} 1 & \text{ if } x \in \mathbb{Q} \\ 0 & \text{ if } x \notin \mathbb{Q}\end{cases}$.
Enter this as
f(x) = d(x)
Other resources: