AP Calculus AB and BC glossary

Properties of definite integrals

Definite integrals obey a small set of rules: swapping the bounds reverses the sign, equal bounds give zero, constants factor out, sums split apart, and an integral can be broken at any interior point.

∫abf(x) dx=∫acf(x) dx+∫cbf(x) dx\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx

The splitting rule holds for any cc, including values outside the interval, which is what lets you combine and rearrange given integrals on table-based free-response questions.

Sign reversal on swapping bounds follows from the Fundamental Theorem: F(b)−F(a)F(b) - F(a) is the negative of F(a)−F(b)F(a) - F(b).

No product rule

There is no rule turning the integral of a product into a product of integrals. That mistake has no analogue that works.

Appears in: Unit 6: Integration and Accumulation