Iterated Integrals

The central idea

An iterated integral evaluates a multiple integral one variable at a time. For a Type I region $D=\{(x,y):a\le x\le b, g_1(x)\le y\le g_2(x)\}$, $\iint_Df\,dA=\int_a^b\int_{g_1(x)}^{g_2(x)}f(x,y)dydx$. The inner bounds may depend on outer variables, but not conversely.

Definitions, hypotheses, and notation

Not every planar region is simple in one orientation; splitting may be unavoidable. To reverse order, identify the region's projection on the new outer axis and determine where left/right or lower/upper boundary formulas change. Algebraic inequalities alone are more error-prone than a labelled sketch.

For rectangles and continuous integrands, bounds are independent and order reversal is immediate. For improper or sign-changing singular integrals, Fubini–Tonelli hypotheses matter: different iterated orders can fail or disagree without absolute integrability.

Conceptual meaning

The inner integral accumulates along one slice while treating outer coordinates as fixed parameters. The outer integral then sums the slice totals. Changing order changes the slicing description, not the geometric region or final value under Fubini hypotheses.

A dependable method and decision rule

1. Sketch the region from inequalities.
2. Read differentials from right to left to identify integration order.
3. Treat outer variables as constants in the inner antiderivative.
4. Substitute inner bounds before performing the outer integral.
5. When reversing order, project and rewrite the region rather than swapping symbols alone.

Fully worked example

Common mistakes and why they fail