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

Graphical or geometric meaning

Vertical slices of the triangle have height $1-x$. As $x$ moves from zero to one, each slice shrinks to a point. Reversing order would use $0\le y\le1$ and $0\le x\le1-y$.

Common mistakes and why they fail

Verification and reasonableness checks

• Set the integrand to one and recover region area.
• Describe each bound in words from the sketch.
• Reverse order and compare when both descriptions are simple.

Inner limits describe a moving slice

In an iterated integral, hold the outer variable fixed while evaluating the inner integral. Variable inner bounds trace the lower and upper edges of the region; outer bounds span the complete projection. Sketching these roles prevents treating a bound expression as a constant. Changing order requires redescribing the same region, not simply swapping differentials. Find intersections and split if a single set of new bounds cannot cover the region without overlap. For continuous functions on a rectangle, either order agrees by Fubini's theorem, providing a check. For nonrectangular regions, set the integrand to one and verify that the result equals geometric area before evaluating a complicated integrand.

Practice

1. Evaluate $\int_0^1\int_0^x1dydx$.
2. Reverse $0\le x\le1$, $x\le y\le1$.
3. What is held constant in an inner $dy$ integral?

Connections and next steps

• Double Integrals
• Triple Integrals
• Double Integrals in Polar Coordinates