Multiple Integrals

Multiple integrals generalize the definite integral to functions of several variables, computing the signed "hyper-volume" under a surface over a region in

R^{n}

. The key computational result is Fubini's Theorem, which reduces an integral over a region to iterated single-variable integrals. They apply to area, volume, mass, center of mass, probability, and flux across mathematics, physics, and engineering. Foundations laid by Cauchy, Riemann, and Lebesgue (19th–20th c.), with iterated integration formalized by Guido Fubini (1907).

A multiple integral is the integral of a function $f : R^{n} \to R$ over a region $D \subseteq R^{n}$ , defined as a limit of Riemann sums over a partition of $D$ into small cells. The most common cases are the double integral $\iint_{D} f d A$ (volume under a surface) and the triple integral $∭_{E} f d V$ (mass, charge, or other distributed quantities in space). Multiple integrals matter because most physical quantities — mass, charge, moment of inertia, gravitational potential, total probability — are accumulated over regions of two or three dimensions, not lines. The single most useful tool is that, under mild conditions, a multiple integral equals an iterated integral: you integrate one variable at a time, treating the others as constants. The surprising fact is that the order of integration can sometimes be exchanged freely (Fubini), yet for improper or non-absolutely-integrable functions, swapping the order can yield different finite answers — a warning that integrability is a genuine hypothesis, not a formality.

The Core Insight

To integrate over a 2D or 3D region, slice it into thin strips or columns, integrate over one variable while holding the others fixed, then integrate the resulting function over the remaining variables. Higher-dimensional accumulation reduces to repeated one-dimensional accumulation — provided the integrand is well-behaved. Geometry enters only through the limits of integration, which encode the shape of the region.

Formal Statement

Double Integral over a Rectangle

Let $R = [a, b] \times [c, d]$ be a rectangle and $f : R \to R$ bounded. Partition $[a, b]$ into subintervals $a = x_{0} < \dots < x_{m} = b$ and $[c, d]$ into $c = y_{0} < \dots < y_{n} = d$ , forming subrectangles $R_{ij}$ of area $Δ A_{ij} = Δ x_{i} Δ y_{j}$ . Choose a sample point $(x_{i}^{*}, y_{j}^{*}) \in R_{ij}$ . The double integral is

\iint_{R} f (x, y) d A = ∥ P ∥ \to 0 lim i = 1 \sum m j = 1 \sum n f (x_{i}^{*}, y_{j}^{*}) Δ A_{ij}

where $∥ P ∥$ is the mesh (largest cell diameter), when this limit exists independent of partition and sample choice.

⟳Mimic the single-variable Riemann integral but partition a 2D region; the product

Δ x_{i} Δ y_{j}

replaces

Δ x_{i}

as the elementary measure.

Here $f (x, y)$ is the integrand, $d A$ the area element, and the value represents the signed volume between the graph $z = f (x, y)$ and the $x y$ -plane over $R$ .

Triple Integral and General $n$-fold Integral

For $f : E \to R$ with $E \subseteq R^{3}$ a bounded region partitioned into cells of volume $Δ V_{ij k}$ ,

∭_{E} f (x, y, z) d V = ∥ P ∥ \to 0 lim i, j, k \sum f (x_{i}^{*}, y_{j}^{*}, z_{k}^{*}) Δ V_{ij k} .

More generally, for $f : D \to R$ , $D \subseteq R^{n}$ ,

\int_{D} f d V_{n} = ∥ P ∥ \to 0 lim k \sum f (x_{k}^{*}) Δ V_{k}, d V_{n} = d x_{1} d x_{2} \dots d x_{n} .

⟳Replace area cells by

n

-dimensional boxes;

Δ V_{k} = \prod_{ℓ = 1}^{n} Δ x_{ℓ}

is the product of side lengths.

Integral over a General (Non-Rectangular) Region

To integrate over a bounded region $D$ , enclose $D$ in a rectangle $R$ and define $\tilde{f} = f$ on $D$ , $\tilde{f} = 0$ off $D$ . Then

\iint_{D} f d A := \iint_{R} \tilde{f} d A,

which exists provided $\partial D$ has zero area (measure zero) and $f$ is bounded and continuous except on a set of measure zero.

⟳Extension by zero only adds cells contributing

0

except near

\partial D

; if the boundary has zero area its contribution vanishes in the limit.

Fubini's Theorem (Continuous Case)

If $f$ is continuous on $R = [a, b] \times [c, d]$ , then the double integral equals either iterated integral:

\iint_{R} f (x, y) d A = \int_{a}^{b} (\int_{c}^{d} f (x, y) d y) d x = \int_{c}^{d} (\int_{a}^{b} f (x, y) d x) d y .

For a type I region $D = {(x, y) : a \leq x \leq b, g_{1} (x) \leq y \leq g_{2} (x)}$ with $g_{1}, g_{2}$ continuous,

\iint_{D} f d A = \int_{a}^{b} \int_{g_{1} (x)}^{g_{2} (x)} f (x, y) d y d x .

⟳For fixed

x

\int_{c}^{d} f d y

is the cross-sectional area

A (x)

; summing

A (x) d x

via the single-variable integral recovers the volume, by uniform continuity controlling the Riemann sum error.

Fubini–Tonelli (Integrability Conditions)

Let $f$ be measurable on $R^{m} \times R^{n}$ . (Tonelli) If $f \geq 0$ , the iterated and double integrals always agree (possibly $+ \infty$ ). (Fubini) If $f$ is integrable, i.e. $\iint ∣ f ∣ d A < \infty$ , then both iterated integrals exist, are equal, and equal $\iint f d A$ .

⟳Tonelli applies to nonnegative functions via the monotone convergence theorem; Fubini decomposes

f = f^{+} - f^{-}

and applies Tonelli to each piece, finiteness of

\iint ∣ f ∣

guaranteeing no

\infty - \infty

Why It Holds / Derivation

Iterated Integration Recovers the Double Integral

We sketch Fubini for continuous $f$ on $R = [a, b] \times [c, d]$ .

Step 1 — Define cross-sectional areas. For each fixed

x \in [a, b]

, the function

y \mapsto f (x, y)

is continuous on

[c, d]

, hence Riemann-integrable. Set

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

. Step 2 — Continuity of $A$ . By uniform continuity of

f

on the compact

R

, given

ε > 0

there is

δ > 0

with

∣ f (x^{'}, y) - f (x, y) ∣ < ε

whenever

∣ x^{'} - x ∣ < δ

. Then

∣ A (x^{'}) - A (x) ∣ \leq ε (d - c)

, so

A

is continuous and integrable. Step 3 — Sandwich the Riemann sums. Partition

R

into cells

R_{ij}

. On each

R_{ij}

let

m_{ij} = in f f

M_{ij} = sup f

. Then

m_{ij} Δ y_{j} \leq \int_{y_{j - 1}}^{y_{j}} f (x_{i}^{*}, y) d y \leq M_{ij} Δ y_{j} .

Summing over

j

gives

\sum_{j} m_{ij} Δ y_{j} \leq A (x_{i}^{*}) \leq \sum_{j} M_{ij} Δ y_{j}

. Step 4 — Multiply by $Δ x_{i}$ and sum over $i$ . This bounds

\sum_{i} A (x_{i}^{*}) Δ x_{i}

between the lower and upper Darboux sums

L (f, P)

and

U (f, P)

f

over

R

. Step 5 — Take the limit. Since

f

is continuous,

U (f, P) - L (f, P) \to 0

∥ P ∥ \to 0

, forcing both to converge to

\iint_{R} f d A

. The squeezed quantity

\sum_{i} A (x_{i}^{*}) Δ x_{i} \to \int_{a}^{b} A (x) d x

. Hence

\int_{a}^{b} A (x) d x = \iint_{R} f d A

. Symmetry in

x, y

gives the other order.

■

Change of Variables (Jacobian Theorem)

Let $T : U \to V$ , $T (u) = x$ , be a $C^{1}$ bijection between open sets with nonvanishing Jacobian determinant. Then

\int_{V} f (x) d V_{x} = \int_{U} f (T (u)) ∣ det D T (u) ∣ d V_{u},

where $D T$ is the Jacobian matrix $\partial x_{i} / \partial u_{j}$ . In 2D with $x = x (u, v), y = y (u, v)$ ,

\iint_{V} f d x d y = \iint_{U} f (x (u, v), y (u, v)) \frac{\partial ( x , y )}{\partial ( u , v )} d u d v, \frac{\partial ( x , y )}{\partial ( u , v )} = det (x_{u} y_{u} x_{v} y_{v}) .

⟳The Jacobian determinant is the local linear volume-scaling factor; a small

uv

-cell of area

d u d v

maps to a parallelogram of area

∣ det D T ∣ d u d v

Polar, Cylindrical, and Spherical Elements

Specializing the change of variables:

Polar: d A = r d r d θ; Cylindrical: d V = r d r d θ d z;

Spherical: d V = ρ^{2} sin ϕ d ρ d ϕ d θ,

with $x = ρ sin ϕ cos θ, y = ρ sin ϕ sin θ, z = ρ cos ϕ$ , $ϕ \in [0, π]$ the polar angle from the $+ z$ axis.

⟳Compute the Jacobian of each coordinate map; e.g. for polar

det (cos θ sin θ - r sin θ r cos θ) = r

Guido Fubini | 1879–1943

An Italian mathematician at Turin, Fubini proved in 1907 the theorem now bearing his name, giving rigorous conditions under which a double (or multiple) integral equals an iterated integral within the framework of Lebesgue measure. His work made the everyday computational practice of "integrate one variable at a time" legitimate beyond the continuous case. Fubini also contributed to differential geometry (the Fubini–Study metric) and projective geometry. A Jewish scholar, he was forced to flee fascist Italy in 1939 and emigrated to the United States, joining the Institute for Advanced Study and Princeton, where he taught until his death.

Building Intuition

Think of $\iint_{D} f d A$ as stacking infinitely thin slabs. Fix $x$ : the slice $z = f (x, y)$ over the segment of $D$ at that $x$ has area $A (x) = \int f (x, y) d y$ . This is a wafer of thickness $d x$ and volume $A (x) d x$ . Sliding $x$ from $a$ to $b$ and summing wafers gives the whole solid: $\int_{a}^{b} A (x) d x$ . The triple integral does the same with one more layer: integrate over $z$ to get a "density per column," then over $y$ to get a "density per slab," then over $x$ . The Jacobian factor in [7] is the conversion rate between coordinate "boxes": a tiny rectangle in $(r, θ)$ -space is not a square in the plane — it is a curved patch of area $r d r d θ$ , fatter far from the origin, which is exactly why the $r$ appears.

Region $D$ as type I, sliced into vertical strips for iterated integration

Worked Examples

Volume Under a Paraboloid Over a Triangle

Given: Find

\iint_{D} (x + 2 y) d A

where

D

is bounded by

y = 2 x^{2}

and

y = 1 + x^{2}

. Step 1 — Find limits. Set

2 x^{2} = 1 + x^{2} \Rightarrow x^{2} = 1 \Rightarrow x = \pm 1

. For

- 1 \leq x \leq 1

, the upper curve is

y = 1 + x^{2}

, lower is

y = 2 x^{2}

(type I). Step 2 — Inner integral over $y$ :

\int_{2 x^{2}}^{1 + x^{2}} (x + 2 y) d y = [x y + y^{2}]_{2 x^{2}}^{1 + x^{2}} .

Upper:

x (1 + x^{2}) + (1 + x^{2})^{2} = x + x^{3} + 1 + 2 x^{2} + x^{4}

. Lower:

x (2 x^{2}) + (2 x^{2})^{2} = 2 x^{3} + 4 x^{4}

. Difference:

- 3 x^{4} - x^{3} + 2 x^{2} + x + 1

. Step 3 — Outer integral over $x$ :

\int_{- 1}^{1} (- 3 x^{4} - x^{3} + 2 x^{2} + x + 1) d x .

Odd terms (

- x^{3}, + x

) vanish. Even terms:

\int_{- 1}^{1} - 3 x^{4} d x = - \frac{6}{5}, \int_{- 1}^{1} 2 x^{2} d x = \frac{4}{3}, \int_{- 1}^{1} 1 d x = 2.

Step 4 — Sum:

- \frac{6}{5} + \frac{4}{3} + 2 = \frac{- 18 + 20 + 30}{15} = \frac{32}{15}

\iint_{D} (x + 2 y) d A = \frac{32}{15} (cubic units)

Mass of a Hemisphere via Spherical Coordinates

Given: A solid hemisphere

ρ \leq R

z \geq 0

, has density

δ (x, y, z) = k z

(kg/m³ per meter,

k

constant). Find its mass. Step 1 — Convert.

z = ρ cos ϕ

d V = ρ^{2} sin ϕ d ρ d ϕ d θ

. Limits:

ρ \in [0, R]

ϕ \in [0, π /2]

θ \in [0, 2 π]

. Step 2 — Set up:

M = \int_{0}^{2 π} \int_{0}^{π /2} \int_{0}^{R} k (ρ cos ϕ) ρ^{2} sin ϕ d ρ d ϕ d θ .

Step 3 — Separate (integrand factors):

M = k (\int_{0}^{2 π} d θ) (\int_{0}^{π /2} cos ϕ sin ϕ d ϕ) (\int_{0}^{R} ρ^{3} d ρ) .

Step 4 — Evaluate each:

\int_{0}^{2 π} d θ = 2 π

;

\int_{0}^{π /2} cos ϕ sin ϕ d ϕ = [\frac{1}{2} sin^{2} ϕ]_{0}^{π /2} = \frac{1}{2}

;

\int_{0}^{R} ρ^{3} d ρ = \frac{R ^{4}}{4}

. Step 5 — Multiply:

M = k \cdot 2 π \cdot \frac{1}{2} \cdot \frac{R ^{4}}{4} = \frac{π k R ^{4}}{4}

M = \frac{π k R ^{4}}{4} kg

Common Misconceptions

$d A \neq = d x d y$ blindly. In polar/spherical coordinates the area/volume element carries the Jacobian: forgetting the $r$ in $r d r d θ$ or $ρ^{2} sin ϕ$ is the single most common error.
Limits are not independent unless the region is rectangular. For non-rectangular $D$ , the inner limits depend on the outer variable; you cannot pull constants out of the inner integral if they are actually functions of the outer variable.
Swapping the order changes the limits, not just the symbols. Converting $\int_{a}^{b} \int_{g_{1} (x)}^{g_{2} (x)} d y d x$ to $d x d y$ order requires re-describing $D$ geometrically, often splitting it into pieces.
Fubini requires integrability. Equality of iterated integrals is not automatic for unbounded or sign-changing integrands.

Fubini Fails Without Absolute Integrability

Let $f (x, y) = \frac{x ^{2} - y ^{2}}{( x ^{2} + y ^{2} ) ^{2}}$ on $(0, 1] \times (0, 1]$ . Note $f = - \partial_{y} (\frac{y}{x ^{2} + y ^{2}})$ , so

\int_{0}^{1} \int_{0}^{1} f d y d x = \int_{0}^{1} \frac{1}{1 + x ^{2}} d x = \frac{π}{4} .

By the antisymmetry

f (x, y) = - f (y, x)

, reversing the order gives

\int_{0}^{1} \int_{0}^{1} f d x d y = - \frac{π}{4} .

The two iterated integrals are both finite but unequal (

π /4 \neq = - π /4

). This is allowed because

\iint ∣ f ∣ d A = + \infty

, so the Fubini hypothesis fails. The double integral itself does not exist.

Connections

Multiple integrals sit at the crossroads of analysis and geometry. The inner integral construction is just the Riemann Integral applied parameterically. The Change of Variables Theorem is the integral analogue of the Jacobian Matrix and Determinant. In 2D and 3D, multiple integrals feed directly into the vector-calculus integral theorems: Green's Theorem equates a double integral of a curl-like quantity to a boundary line integral, while the Divergence Theorem equates a triple integral of $\nabla \cdot F$ to a flux Surface Integral. The rigorous foundation generalizes to Lebesgue Integration, where Fubini–Tonelli reaches its full strength, and to Differential Forms, under which $d x \land d y$ encodes orientation and the change-of-variables Jacobian appears automatically. In probability, the joint-density integral is the gateway to Marginal and Conditional Distributions.

Real-World Applications

Center of mass / centroid. $\overset{x}{ˉ} = \frac{1}{M} ∭_{E} x δ d V$ designs balanced structures and aircraft (load and balance).
Moment of inertia. $I = ∭_{E} r_{⊥}^{2} δ d V$ governs rotational dynamics of flywheels, turbines, and spacecraft attitude control.
Electromagnetism. Total charge $Q = ∭_{E} ρ d V$ and electrostatic potential $V (r) = \frac{1}{4 π ε _{0}} ∭ \frac{ρ ( r ^{'} )}{∣ r - r ^{'} ∣} d V^{'}$ .
Probability. For a joint density $f_{X Y}$ , $P ((X, Y) \in A) = \iint_{A} f_{X Y} d A$ ; normalization requires $\iint_{R^{2}} f = 1$ .
Fluid mechanics / heat transfer. Mass flow rate and total thermal energy $∭ ρ c T d V$ in reactors and HVAC design.
Gravitation and geophysics. Newtonian gravitational field of an extended body via triple integration of mass density; used in satellite geodesy and gravity surveys for oil/mineral exploration.
Computer graphics & rendering. The rendering equation evaluates radiance as integrals over hemispheres of incoming light; Monte Carlo estimation of these multiple integrals underlies ray tracing.

Historical Development

1731 — Euler uses iterated integration of functions of two variables in early studies, treating areas/volumes via repeated antidifferentiation.
1769 — Euler publishes systematic treatment of double integrals and change of variables (the substitution that prefigures the Jacobian).
1813 — Gauss develops surface and volume integrals while working on potential theory and the attraction of ellipsoids.
1823 — Cauchy gives the first rigorous definition of the (single) integral as a limit of sums, the template later extended to higher dimensions.
1841 — Jacobi formalizes the determinant (the "Jacobian") governing transformation of multiple integrals under coordinate change.
1854 — Riemann defines the Riemann integral rigorously via upper/lower sums, directly generalizable to $R^{n}$ .
1902 — Lebesgue introduces measure-theoretic integration, enabling integration of far broader classes of functions and clean multidimensional theory.
1907 — Fubini proves the iterated-integral theorem in the Lebesgue setting.
1909 — Tonelli supplies the nonnegative-function version, completing the Fubini–Tonelli framework still used today.

Subtleties & Limits

Multiple integration is computationally settled in low dimensions but raises deep practical and theoretical issues. The curse of dimensionality: deterministic quadrature with $k$ nodes per axis needs $k^{n}$ evaluations, which is infeasible for large $n$ (e.g., path integrals in physics or high-dimensional Bayesian inference). This motivates Monte Carlo and quasi-Monte Carlo methods, whose error analysis (effective dimension, discrepancy bounds) is an active area. The general $n$ -dimensional change of variables for non-smooth or non-injective maps requires degree theory and the coarea/area formulas of geometric measure theory. For oscillatory integrands $\int f e^{iλ ϕ} d V$ , sharp decay rates as $λ \to \infty$ tie into open questions in harmonic analysis (the restriction and Kakeya problems). Even the seemingly elementary question of when one may interchange a limit/derivative with a multiple integral demands dominated-convergence hypotheses that are routinely, and sometimes wrongly, assumed.