To measure the area under a graph, cut the interval into pieces and on each piece take a rectangle that is certainly too tall and one that is certainly too short. Adding them gives a number above the answer and a number below it. Refining the cut can only lower the first and raise the second, so the two families of numbers press towards each other — and whether they meet is the question the whole chapter is about.

A field of irregular shape is measured by covering it with square tiles: count the tiles that overlap it at all, and the tiles entirely inside. The first count is too many and the second too few, and using smaller tiles narrows the gap. The area is what the two counts agree on, if they ever do.

A partition of $[a,b]$ is a finite list of points starting at $a$ and ending at $b$. The pieces need not have equal widths, and nothing in the definition asks them to.

A partition $P$ of $[a,b]$ is a finite set $a=x_0<x_1<\dots<x_n=b$. For a bounded put and . The upper sum is and the lower sum , where . Boundedness is needed for the suprema and infima to exist, and it is assumed from here on.

• $L(f,P)\le U(f,P)$ for every partition, since $m_i\le M_i$ on each piece.

It is enough to add one point, since a refinement is reached by adding points one at a time. Adding a point splits one piece into two. The supremum over each half is at most the supremum over the whole, because the half is a subset, so the two new rectangles together are no taller than the old one over the same total width. The upper sum therefore does not rise; the argument for infima is the mirror image, and the middle inequality is the one already known.

It is enough to add a single point, and repeat.

A supremum over a subset is at most the supremum over the whole piece.

The two new widths add to the old one.

So the two new rectangles contribute at most what the old one did, and nothing else changes.

The mirror argument with infima, which are at least the old infimum on each half.

A partition is a finite increasing list from one endpoint to the other. Equal widths are a convenience and never a requirement.

Two unrelated partitions are compared through the partition containing both.

• A partition is a finite increasing list from one endpoint to the other.
• Upper and lower sums use the supremum and infimum on each piece.
• Refining lowers the upper sum and raises the lower sum.
• Every lower sum is below every upper sum, by the common refinement.

Great work! You now know how to:

• Write down a partition and compute an upper and a lower sum
• Say which way each sum moves under refinement
• Prove that every lower sum is below every upper sum
• Say why boundedness is assumed throughout