Double integrals over rectangles — Fubini's theorem: Foundations

Before you can read the parent note Double integrals over rectangles — Fubini's theorem, you need every symbol it silently assumes. This page builds each one from nothing, in the order they stack. Nothing here uses a symbol before it is drawn.

1. Points and coordinates: $(x, y)$ and the plane

Why the topic needs it. The floor we integrate over lives in this plane. Every "sample point" $(x_{i}^{*}, y_{j}^{*})$ in the parent's Riemann sum is just one dot here. If $(x, y)$ isn't crystal clear, the whole sum is fog.

2. A function of two variables: $z = f (x, y)$

Picture a floor with a fabric stretched above it. Over each floor-point the fabric sits at some height — that height is $f (x, y)$ . As you slide $(x, y)$ around, the fabric rises and dips: a surface.

Why the topic needs it. The parent's phrase "surface $z = f (x, y)$ " and "volume under the surface" only make sense once you see $f$ as a roof height over each floor-point. If $f (x, y)$ is negative, the roof dips below the floor — remember this for signed volume later.

Recall Test yourself

If $f (x, y) = x + 2 y$ , what is the height above the point $(1, 3)$ ? ::: $1 + 2 (3) = 7$ .

3. The rectangle $R = [a, b] \times [c, d]$

In set language the parent writes $R = [a, b] \times [c, d] = {(x, y) : a \leq x \leq b, c \leq y \leq d} .$

Read it aloud: "the set of points $(x, y)$ such that $x$ sits between $a$ and $b$ AND $y$ sits between $c$ and $d$ ." The colon $:$ means "such that"; the comma means "and."

Why the topic needs it. $R$ is the floor we build volume over. Because its edges are fixed numbers $a, b, c, d$ (not curved, not depending on each other), the integration limits will be plain constants — which is exactly why the rectangular Fubini theorem is the easy first case, before Double integrals over general regions.

4. Chopping it up: $Δ x$ , $Δ y$ , $Δ A$ , and the partition $P$

Slice $[a, b]$ into thin strips of width $Δ x$ and $[c, d]$ into thin strips of height $Δ y$ . The grid lines carve $R$ into tiny sub-rectangles. This whole grid is called a partition $P$ .

Why the topic needs it. A tower of base $Δ A$ and height $f (x_{i}^{*}, y_{j}^{*})$ has volume $f \cdot Δ A$ . Shrinking $∥ P ∥$ turns the staircase of towers into the smooth true volume. This is the heartbeat of the definition — and it's the 2-D twin of Riemann sums.

5. Sample point $(x_{i}^{}, y_{j}^{})$ and the star $*$

Why the topic needs it. We can't use a whole tile's worth of heights, so we pick one height $f (x_{i}^{*}, y_{j}^{*})$ to stand for the tower on that tile. Once tiles shrink to dots, the choice stops mattering — every point in a dot-sized tile has nearly the same height.

6. The summation signs $i \sum j \sum$

So the parent's Riemann sum $S = \sum_{i} \sum_{j} f (x_{i}^{*}, y_{j}^{*}) Δ A$ reads: "add the volume of every little tower over every tile."

Why the topic needs it. This finite sum is the approximation. The integral is its limit. Notice the double sum already hints at Fubini: adding across-then-up equals adding up-then-across — order of the two sigmas doesn't change the total.

7. The integral signs: $\int$ , $\iint$ , and $d A$

8. The iterated integral: $\int_{a}^{b} \int_{c}^{d} f d y d x$

Why the topic needs it. Computing the 2-D limit directly is brutal. The iterated integral turns one hard 2-D problem into two easy 1-D integrals you already know how to do. That is the entire practical payoff of Fubini — and once you can swap the order, you unlock Change of order of integration.

9. Prerequisite map

Read it as: points and functions build a surface over a rectangle; chopping the rectangle builds the Riemann sum; its limit is the double integral; slicing turns that into iterated integrals; matching the two slicing directions is Fubini.

Equipment checklist

Say each answer out loud before revealing.

What does the point

(x, y)

mean, and which coordinate is horizontal?

A spot on the plane;

x

is horizontal (across),

y

is vertical (up).

What does

z = f (x, y)

represent geometrically?

The height of a surface (roof) above the floor-point

(x, y)

Write the rectangle

R = [a, b] \times [c, d]

in set-builder words.

All points

(x, y)

such that

a \leq x \leq b

and

c \leq y \leq d

What is

Δ A

and how is it built?

The area of one small tile,

Δ A = Δ x Δ y

(width times height).

What does

∥ P ∥ \to 0

mean physically?

The biggest tile shrinks to a dot; the whole grid becomes infinitely fine.

What does the star in

(x_{i}^{*}, y_{j}^{*})

flag?

A chosen sample point inside a tile where we measure the roof height.

Translate

\sum_{i} \sum_{j} f (x_{i}^{*}, y_{j}^{*}) Δ A

into one sentence.

Add up the volume of every tiny tower over every tile.

What is the difference between

d A

and

d x d y

d A

is a direction-free patch of area;

d x d y

commits to an integration order — on a rectangle they are equal.

\int_{a}^{b} \int_{c}^{d} f d y d x

, which integral runs first and what is frozen?

The inner

d y

-integral runs first;

x

is frozen as a constant.

What does

A (x) = \int_{c}^{d} f (x, y) d y

represent in the loaf-of-bread picture?

The area of one slice's face at position

x

Connections

Riemann sums — the double sum here is the 2-D version.
Volume by slicing (single-variable) — where $A (x)$ and $\int A (x) d x$ come from.
Double integrals over rectangles — Fubini's theorem — the parent this page equips you for.
Double integrals over general regions — next, when the edges of $R$ curve.
Change of order of integration — swapping $d x d y$ once you trust Fubini.
Fubini–Tonelli theorem — the deeper "when is swapping allowed" statement.
Triple integrals — one more dimension, tiny boxes instead of tiles.