AP Calculus AB and BC glossary

Definite integral

A definite integral is a number defined as the limit of Riemann sums as the subintervals shrink to zero width. It represents the net signed accumulation of a quantity between two bounds, with area below the axis counting as negative.

∫abf(x) dx=lim⁡n→∞∑i=1nf(xi∗) Δx\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*)\,\Delta x

The word signed is doing real work. Region below the horizontal axis contributes negatively, so a definite integral can be zero or negative even though it is often described as area. For genuine area you integrate the absolute value.

The Fundamental Theorem of Calculus is what makes these computable without ever forming a Riemann sum, by connecting the definite integral to any antiderivative of the integrand.

It is a number

The variable of integration is a placeholder, so ∫abf(x) dx\int_a^b f(x)\,dx and ∫abf(t) dt\int_a^b f(t)\,dt are the same number. Nothing in the answer depends on the letter used.

Appears in: Unit 6: Integration and Accumulation