upper Darboux sum
An upper Darboux sum is the most generous staircase you can build over a partition: on each strip you make the rectangle as tall as the function ever gets there, so the staircase always sits on top of the graph. It deliberately over-estimates the area, giving a ceiling you can trust never to fall below the truth.
Let f be bounded on [a, b] and let P = {x_0, ..., x_n} be a partition. On the k-th subinterval set M_k = sup{ f(x) : x_(k-1) <= x <= x_k }. The upper Darboux sum is U(f, P) = sum_{k=1}^{n} M_k (x_k - x_(k-1)). Because f is bounded, each M_k is a finite real number, so U(f, P) is well defined.
Unlike a Riemann sum, the upper sum involves no choice of tag: the supremum over each subinterval is forced. Consequently for any tagged partition with the same P, the Riemann sum never exceeds U(f, P), and U(f, P) is itself the supremum of those Riemann sums. Refining a partition can only lower an upper sum, which is why the upper integral is defined as an infimum over all partitions.
For f(x) = x on [0, 1] with the uniform partition into n pieces, M_k = k/n (the right endpoint is the sup), so U(f, P) = sum (k/n)(1/n) = (n+1)/(2n), which decreases to 1/2.
For an increasing function the supremum on each piece is the right endpoint, so the upper sum approaches the integral from above.
Boundedness is essential: if f is unbounded on some subinterval then M_k = +infinity and the Darboux machinery breaks down, which is why the Riemann and Darboux integrals are defined only for bounded functions on bounded intervals.