tagged partition
A plain partition tells you where to cut the interval but leaves the height of each approximating rectangle undecided. A tagged partition fixes that gap: in every subinterval you pick one sample point — a tag — and the function's value there will become the rectangle's height. Think of dropping a single pin somewhere inside each strip and reading off the curve at the pin.
Formally, a tagged partition of [a, b] is a pair (P, T) where P = {x_0, ..., x_n} is a partition and T = (t_1, ..., t_n) is a choice of tags with t_k lying in the k-th subinterval, x_(k-1) <= t_k <= x_k. The tag is allowed to be any point of its closed subinterval, including an endpoint, and different subintervals are tagged independently.
Tagged partitions are precisely the data a Riemann sum needs: sum f(t_k)(x_k - x_(k-1)). Because the tags are free, the same underlying partition can produce many different Riemann sums; integrability is the statement that all these choices are forced together as the mesh shrinks. The Darboux approach sidesteps the choice of tags by using suprema and infima instead, and the two approaches turn out to be equivalent.
On [0, 1] partitioned into n equal pieces, choosing the right endpoint t_k = k/n as each tag gives the right-endpoint Riemann sum; choosing the midpoint t_k = (2k-1)/(2n) gives the midpoint rule.
Same partition, different tags, different sums — yet for an integrable function they all converge to one number.
Tagged partitions are the natural setting for the gauge (Henstock–Kurzweil) integral, where one allows the allowed mesh to depend on the tag. In the ordinary Riemann theory the tag is just a free choice within a fixed-width subinterval.