Iterated integral (${\int }_a^b\left({\int }_{c(y)}^{d(y)}f(x,y)dx\right)dy$)

An iterated integral performs successive one-variable integrals. In the displayed expression, $y$ is held fixed during the inner integration over $x$, and the result is then integrated over $y$. The inner bounds may depend on that fixed outer variable. By contrast, an inner bound depending on an already integrated variable does not specify such a nested operation without further interpretation. For example, the triangle $0\le y\le 1$, $2y\le x\le 2$ gives ${\int }_0^1{\int }_{2y}^{2}f(x,y)dxdy$, or, in the other order, ${\int }_0^2{\int }_0^{x/2}f(x,y)dydx$. Fubini's theorem and Tonelli's theorem supply conditions under which these iterated expressions equal the same multiple integral.

The sequence in which variables are integrated in an iterated integral. Reversing the order changes the slicing of the domain of integration, often requiring more than one integral. Fubini's theorem permits interchange for absolutely integrable functions, while Tonelli's theorem permits nonnegative measurable integrands even when the integral is infinite. Conditional convergence requires additional care; geometric reversal alone is not a general justification for interchanging improper integrals.