AP Calculus AB and BC glossary

Variable of integration

Also called: Integration variable

The variable of integration is the letter named by the differential, and it tells you what you are integrating with respect to and which other letters count as constants. In a definite integral it is a dummy name, so renaming it leaves the value unchanged.

In a definite integral the name is disposable. $\int_0^3 t^2\,dt$ and $\int_0^3 u^2\,du$ are both the number 9, and the letter never appears in the answer. An indefinite integral is different, because there the name survives: $\int x^2\,dx = \frac{x^3}{3} + C$ .

The differential also decides what is being held constant. Treating $t$ as the variable gives $\int e^{kt}\,dt = \frac{1}{k}e^{kt} + C$ for $k \neq 0$ , while treating $k$ as the variable gives $\int e^{kt}\,dk = \frac{1}{t}e^{kt} + C$ for $t \neq 0$ . Same expression, two different problems.

This is why accumulation functions use a second letter. In $g(x) = \int_1^x f(t)\,dt$ the $t$ runs across the interval while $x$ is the input to $g$ , so the two roles need two names.

The mistake

Changing the function but not the differential. Once $u = x^2$ and $du = 2x\,dx$ , the integral $\int 2x\cos(x^2)\,dx$ becomes $\int \cos u\,du$ , never $\int \cos u\,dx$ . A leftover $dx$ next to a $u$ means the substitution is unfinished.

Appears in: Unit 6: Integration and Accumulation