Topic 6.6 Applying Properties of Definite Integrals: use linearity, additivity over intervals, and limit-reversal properties of definite integrals.

What this topic is asking

The College Board (Topic 6.6) gives the algebraic properties of definite integrals: linearity (constants factor out, sums split), additivity over adjacent intervals, the zero-width property, and the sign flip when limits are reversed. You use these to combine given integral values without ever evaluating an integral directly.

The properties

A worked combination

Why additivity and reversal matter

Interval additivity is the property that lets you split an integral at a convenient interior point, or fill a gap when you know the integral over two pieces but want the whole. Reversal handles integrals written "backward", with the larger number on the bottom, which arise when a problem gives $\int_5^1 f$; flipping to $\int_1^5 f$ with a sign change makes it usable. Together with linearity, these three properties answer almost every "given these integral values, find this one" question, which is a staple of the no-calculator section. None of them requires knowing the function $f$ itself.

Reading the properties off geometry

Each property has a geometric meaning that makes it easy to remember. Linearity reflects that scaling a function scales its signed area and that adding functions adds their areas. Additivity reflects that the area from $a$ to $c$ is the area from $a$ to $b$ plus the area from $b$ to $c$, for an interior point $b$. Reversal reflects the convention that sweeping the interval backward negates the signed area. The zero-width property reflects that an interval of no width encloses no area. Anchoring the algebra to these pictures prevents sign errors, especially the common mistake of forgetting the sign change when limits are reversed.

Combining the properties in one problem

Exam questions often require several properties at once. A typical no-calculator item gives a few integral values and asks you to find another that requires reversing one integral, splitting another, factoring out a constant, and adding. The strategy is to write the target integral in terms of the given ones using additivity to match intervals, reversal to fix any backward limits, and linearity to handle constants and sums, then substitute the known numbers. Working step by step, transforming the target until every piece is a given value, keeps a multi-property problem from becoming confusing. Each property is simple on its own; the difficulty is only in chaining them, so do the transformations one at a time.

Using symmetry as an extra tool

A related shortcut, though strictly a special case rather than one of the core properties, is symmetry. For an even function ($f(-x) = f(x)$), $\int_{-a}^{a} f = 2\int_0^a f$, since the signed areas on the two halves match. For an odd function ($f(-x) = -f(x)$), $\int_{-a}^{a} f = 0$, because the halves cancel. Recognizing even or odd structure can turn a hard integral into a trivial one on a symmetric interval, and it explains why integrals like $\int_{-1}^{1}(x^3 - x)\,dx$ come out to zero. While the AB exam states symmetry as a fact you may use, it rests on the same additivity and reversal ideas applied to the two halves of a symmetric interval.