JOVANA
Library Glossary Getting Started Three Levels Fields How it works Mission
Join the mission
All guides

Cutting Up the Interval: Partitions and Darboux Sums

Before we can integrate we must approximate. We chop [a,b] into pieces, trap the area between sums that are too big and too small, and watch the gap shrink.

What a partition is

Fix a bounded function f on a closed interval [a,b]. A partition P of [a,b] is just a finite list of points a = x_0 < x_1 < … < x_n = b. These points cut the interval into n subintervals [x_{k-1}, x_k]. The whole game of integration is to approximate the area under f by adding up the areas of thin rectangles, one over each subinterval — and to make the approximation as good as we like by cutting finely enough.

The width of subinterval k is Δx_k = x_k − x_{k-1}. The largest of these widths is the mesh of P, written ‖P‖ = max_k Δx_k. A small mesh means every piece is thin. We do not require the pieces to be equal; allowing uneven pieces makes some proofs cleaner.

Upper and lower sums

On each subinterval, how tall should our rectangle be? Two honest choices: the tallest the function gets and the shortest it gets. Because f is bounded, on subinterval k it has a supremum M_k = sup f over [x_{k-1}, x_k] and an infimum m_k = inf f over the same piece. The upper Darboux sum uses the tall rectangles and the lower Darboux sum uses the short ones:

U(f, P) = sum_{k=1}^{n} M_k * Δx_k   with M_k = sup f on [x_{k-1}, x_k]
L(f, P) = sum_{k=1}^{n} m_k * Δx_k   with m_k = inf f on [x_{k-1}, x_k]

Since m_k <= f(t) <= M_k for every t in the k-th piece,
   L(f, P) <= (true area, if it exists) <= U(f, P)   for EVERY partition P.

Example: f(x) = x^2 on [0, 1], partition P = {0, 1/2, 1}, Δx_k = 1/2.
  Piece [0, 1/2]: m_1 = 0,    M_1 = 1/4
  Piece [1/2, 1]: m_2 = 1/4,  M_2 = 1
  L(f, P) = 0*(1/2) + (1/4)*(1/2) = 1/8 = 0.125
  U(f, P) = (1/4)*(1/2) + 1*(1/2) = 5/8 = 0.625
So the area lies in [0.125, 0.625]. (The true value is 1/3 = 0.333…)
Trapping the area of x^2 on [0,1] between a lower and an upper sum.

Refining makes things better, never worse

A partition Q is a refinement of P if Q contains every point of P plus possibly more. The key structural fact: refining can only raise the lower sum and lower the upper sum. Adding a cut point splits one piece into two; on each smaller piece the inf can only be larger and the sup can only be smaller, because you are taking sup/inf over a smaller set.

  1. If Q refines P, then L(f, P) ≤ L(f, Q) ≤ U(f, Q) ≤ U(f, P). The lower sum climbs, the upper sum drops, and they keep sandwiching the area.
  2. Given ANY two partitions P_1 and P_2, their common refinement Q = P_1 ∪ P_2 satisfies L(f, P_1) ≤ L(f, Q) ≤ U(f, Q) ≤ U(f, P_2).
  3. Therefore EVERY lower sum ≤ EVERY upper sum — even sums from totally unrelated partitions. The set of lower sums sits entirely below the set of upper sums.