Changing order of integration: Foundations — Maths deep dive

This page builds every symbol the parent note "Changing order of integration" leans on. If a symbol below is new to you, read its three parts: plain words → the picture → why we need it. Nothing is used before it is built.

1. The coordinate plane: $x$ , $y$ , and a point $(x, y)$

The picture: two rulers glued at right angles, each ruler running through negative numbers on one side of the crossing point and positive on the other. Every dot in the flat sheet — left or right, high or low — has exactly one address $(x, y)$ .

Why the topic needs it: the whole subject is about a flat region, and a region is nothing but "which addresses $(x, y)$ are inside." A region can sit anywhere — even straddling the axes into negative $x$ or $y$ — so the address system must handle every sign. Every limit we write is a rule filtering addresses.

2. Curves as equations: $y = g (x)$ and $x = h (y)$

The picture: a wiggly line drawn on the plane. Reading it as $y = g (x)$ means scanning left-to-right; reading the same line as $x = h (y)$ means scanning bottom-to-top.

3. Inequalities carve out a region: $a \leq x \leq b$

The picture: the four inequalities are four fences. The region $R$ is the yard they enclose.

Why the topic needs it: limits of an integral are these inequalities in disguise. The 5-step recipe in the parent note is "turn limits into inequalities, sketch, re-read the inequalities the other way."

4. Type I vs Type II: the strip you sweep with

The picture: the same triangle, once shredded into vertical matchsticks, once into horizontal matchsticks. Same triangle, different matchsticks.

Why the topic needs it: "changing the order" = "switching from vertical matchsticks to horizontal ones." The number of dust specks on the whole patio is the same either way — that is the deep reason the answer never changes.

5. What the integral sign actually asks: $\int$ , $\iint$ , $d A$

The symbols:

$\int$ = an "S" for Sum, stretched tall.
$\iint$ = two sums because we cover a 2D region (two directions to sweep).
$d A$ = the shrunk tile of Area; in coordinates $d A = d x d y = d y d x$ .
$f (x, y)$ = the number each address carries.

Why the topic needs it: because a sum can be totalled rows-then-columns or columns-then-rows and give the same total, both iterated integrals equal the one $\iint_{R} f d A$ .

Formula When exactly is the reorder legal? — Fubini's hypotheses

Swapping the order is not automatic for every function. Fubini's Theorem guarantees $\int \int f d y d x = \int \int f d x d y = \iint_{R} f d A$ provided one of these holds:

$f$ is continuous on the closed bounded region $R$ (the case in every example on this site), or
$f$ is nonnegative on $R$ (Tonelli's version — the total is the same even if it is $+ \infty$ ), or
$\iint_{R} ∣ f ∣ d A < \infty$ (the total of the absolute value is finite — "absolutely integrable").

Why the caution matters: for wildly oscillating or unbounded $f$ where none of these hold, the two orders can give different numbers. So "the answer never changes" is a promise that applies to well-behaved $f$ — check a hypothesis before you trust the swap, especially with improper integrals (Improper and Non-elementary Integrals).

6. The one tool that makes reordering worth it: antiderivatives

Why the topic needs it: if the inner integral has no closed form in one order, flipping may hand you an inner integral that does. That is motivation #1 in the parent note.

Worked example The magic cancellation, shown in full context

Take the whole iterated integral over the unit triangle in $d x d y$ order: $\int_{y = 0}^{1} (\int_{x = 0}^{y} \frac{s i n y}{y} d x) d y .$ Focus on the inner integral. Here $x$ is the dummy variable — it is the letter that moves across the strip and disappears after integration. The quantity $\frac{s i n y}{y}$ contains no $x$ , so relative to the inner $\int d x$ it is a constant and simply pulls out. The upper limit $y$ is the strip length — it depends on the outer variable $y$ , which is held fixed while we do the $x$ -integral: $\int_{x = 0}^{y} \frac{s i n y}{y} d x = \frac{s i n y}{y} [x]_{x = 0}^{x = y} = \frac{s i n y}{y} \cdot y = sin y .$ The dummy $x$ vanished; the outer-dependent limit $y$ cancelled the awkward $\frac{1}{y}$ . What remains is the outer integral $\int_{0}^{1} sin y d y$ , which is elementary. That cancellation is the entire payoff of reordering.

Prerequisite map

Read the diagram below top to bottom: each box is a foundation from this page, and every arrow means "you need the upper idea before the lower one makes sense." Trace any path down to the bottom box "Changing order of integration" — that path is a valid study order. If a box feels shaky, jump back to its section above before continuing. (If the diagram fails to render in your reader, the same dependencies are spelled out in the Equipment checklist questions that follow.)

Equipment checklist

Test yourself — reveal only after you've answered aloud.

What does the address

(x, y)

mean, including negative numbers?

Move

x

sideways (left if

x < 0

, right if

x > 0

) then

y

vertically (down if

y < 0

, up if

y > 0

) to land on one dot.

Give the two names of the line through the origin at

4 5^{\circ}

y = x

and, solved the other way,

x = y

— the same curve.

Solve

y = x^{2}

for

x

on a region of positive

x

x = + y

(positive branch only).

Why can't a full circle

x^{2} + y^{2} = 1

be written as a single

y = g (x)

Each

x

gives two

y

-values (upper and lower arcs); you must split it into

y = + 1 - x^{2}

and

y = - 1 - x^{2}

or split the region.

\int_{a}^{b} \int_{g_{1}}^{g_{2}} f d y d x

, which limits must be constants?

The outer ones

a, b

; the inner may depend on

x

A vertical strip corresponds to which differential order?

d y d x

(inner

y

up the strip, outer

x

steps sideways).

What does

\iint_{R} f d A

literally compute?

The limit of

\sum f Δ x Δ y

— the total of

f

over all tiny tiles of

R

Name one hypothesis under which Fubini lets you swap the order.

f

continuous on closed bounded

R

(or

f \geq 0

, or

\iint ∣ f ∣ < \infty

Why bother flipping the order at all?

The inner integral may have no elementary antiderivative one way but a trivial one the other way.

When you flip a strip

9 0^{\circ}

, what must you do to each boundary curve?

Re-solve it for the new inner variable (swap

g

for

h

Next: with these symbols secured, the parent note Changing order of integration turns them into the 5-step reorder recipe. Related deep tools live in Double Integrals over General Regions, Polar Coordinates in Double Integrals and Change of Variables (Jacobian).