What Splits, and What Does Not

You'll learnto rearrange a definite integral without evaluating it — splitting at a point, reversing the limits, pulling a constant out, breaking a sum apart — and to recognise the two rearrangements that look exactly like the others and are not rules at all.

A short list of rules lets you rearrange definite integrals without evaluating anything, and the list is short on purpose. Sums and constant multiples come apart. Products and quotients never do. Splitting a region at a point is free, reversing the direction of travel costs a minus sign, and on a centred interval symmetry can do half the work or all of it. Everything here is one picture: a region, a direction, and a number.

8 STEPS · 6 QUICK CHECKS · SUMS AND CONSTANT MULTIPLES SPLIT · PRODUCTS NEVER DO · v1

a region, a direction of travel, and a number that can be rearranged but not invented

Before you start

What you're looking at

An integrand on top with the region between it and the axis shaded — accent where it counts positively, violet where it counts negatively. Underneath, for most of the file, a ledger: each definite integral drawn as an arrow running from its lower limit to its upper limit on the same horizontal scale.

The question

Which rearrangements of a definite integral are allowed, and which two look identical to those and are not rules at all?

Watch for

The arrow's direction. Reversing the limits does not move the region by a pixel; it turns the arrow round, and the sign is what records that. And in step 6, two numbers on screen that disagree — that is a counterexample, not a caption.

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