Applying Properties of Definite Integrals

A definite integral measures net signed area, built up from pieces f(x) times a tiny width dx, added across an interval. That construction explains two rules you'll use constantly. First, the integral from a to b equals the negative of the integral from b to a. Reversing the limits reverses every width dx to a negative width, since you're now walking backward along the x-axis, so every sliver in the sum flips sign and the total flips sign with it. Second, for any point c between a and b, the integral from a to b splits as the integral from a to c plus the integral from c to b. This is just chopping the same stack of slivers into two groups and adding them back up — the areas combine exactly like lengths on a number line.

These rules also work in reverse: given pieces, you can assemble or rearrange them to get a wanted integral, even reversing one piece's limits to flip its sign before adding. What integrals do NOT do is distribute over a product. Sum rule, difference rule, and constant multiple rule all hold because sums of sums rearrange freely. But the integral of f(x)g(x) is a limit of sums of f(x)g(x) dx, and there's no way to peel that apart into [sum of f dx] times [sum of g dx] — products of sums don't equal sums of products, term by term.

These same splitting and sign rules become essential once the Fundamental Theorem lets you evaluate integrals using antiderivatives: you'll split integrals at points where a function's formula changes, or where it has a corner, and you'll flip limits to match an antiderivative's natural setup. Mastering this bookkeeping now makes every Fundamental Theorem computation afterward go smoothly.