Multiple Integrals

⚡ Key Takeaways

Multiple integrals generalize the definite integral to functions of several variables, computing the signed "hyper-volume" under a surface over a region in $\mathbb{R}^n$ . 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).

💡 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.

DefinitionDouble Integral over a Rectangle

Let $R=[a,b]\times[c,d]$ be a rectangle and $f:R\to\mathbb{R}$ 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 $\Delta A_{ij}=\Delta x_i\,\Delta y_j$ . Choose a sample point $(x_i^*,y_j^*)\in R_{ij}$ . The double integral is

1

\quad \iint_R f(x,y)\,dA \;=\; \lim_{\|P\|\to 0}\;\sum_{i=1}^{m}\sum_{j=1}^{n} f(x_i^*,y_j^*)\,\Delta A_{ij}

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

DefinitionTriple Integral and General

n

-fold Integral

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

2

\quad \iiint_E f(x,y,z)\,dV \;=\; \lim_{\|P\|\to 0}\sum_{i,j,k} f(x_i^*,y_j^*,z_k^*)\,\Delta V_{ijk}.

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

3

\quad \int_D f\,dV_n \;=\; \lim_{\|P\|\to0}\sum_k f(\mathbf{x}_k^*)\,\Delta V_k,\qquad dV_n = dx_1\,dx_2\cdots dx_n.

DefinitionIntegral 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

4

\quad \iint_D f\,dA \;:=\; \iint_R \tilde f\,dA,

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

TheoremFubini's Theorem (Continuous Case)

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

5

\quad \iint_R f(x,y)\,dA \;=\; \int_a^b\!\!\left(\int_c^d f(x,y)\,dy\right)dx \;=\; \int_c^d\!\!\left(\int_a^b f(x,y)\,dx\right)dy.

For a type I region $D=\{(x,y): a\le x\le b,\; g_1(x)\le y\le g_2(x)\}$ with $g_1,g_2$ continuous,

6

\quad \iint_D f\,dA \;=\; \int_a^b\!\int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx.

TheoremFubini–Tonelli (Integrability Conditions)

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

ProofIterated 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 $A(x)=\int_c^d f(x,y)\,dy$ .

Step 3 — Sandwich the Riemann sums. Partition $R$ into cells $R_{ij}$ . On each $R_{ij}$ let $m_{ij}=\inf f$ , $M_{ij}=\sup f$ . Then

m_{ij}\,\Delta y_j \le \int_{y_{j-1}}^{y_j} f(x_i^*,y)\,dy \le M_{ij}\,\Delta y_j.

Summing over $j$ gives $\sum_j m_{ij}\Delta y_j \le A(x_i^*) \le \sum_j M_{ij}\Delta y_j$ .

Step 4 — Multiply by $\Delta x_i$ and sum over $i$ . This bounds $\sum_i A(x_i^*)\Delta x_i$ between the lower and upper Darboux sums $L(f,P)$ and $U(f,P)$ of $f$ over $R$ .

Step 5 — Take the limit. Since $f$ is continuous, $U(f,P)-L(f,P)\to 0$ as $\|P\|\to0$ , forcing both to converge to $\iint_R f\,dA$ . The squeezed quantity $\sum_i A(x_i^*)\Delta x_i\to \int_a^b A(x)\,dx$ . Hence $\int_a^b A(x)\,dx=\iint_R f\,dA$ . Symmetry in $x,y$ gives the other order. $\blacksquare$

∎

TheoremChange of Variables (Jacobian Theorem)

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

7

\quad \int_{V} f(\mathbf{x})\,dV_{\mathbf x} \;=\; \int_{U} f(T(\mathbf{u}))\,\left|\det DT(\mathbf{u})\right|\,dV_{\mathbf u},

where $DT$ is the Jacobian matrix $\partial x_i/\partial u_j$ . In 2D with $x=x(u,v),\,y=y(u,v)$ ,

8

\quad \iint_V f\,dx\,dy = \iint_U f\bigl(x(u,v),y(u,v)\bigr)\,\left|\frac{\partial(x,y)}{\partial(u,v)}\right|du\,dv,\quad \frac{\partial(x,y)}{\partial(u,v)}=\det\!\begin{pmatrix} x_u & x_v\\ y_u & y_v\end{pmatrix}.

DefinitionPolar, Cylindrical, and Spherical Elements

Specializing the change of variables:

9

\quad \text{Polar: } dA = r\,dr\,d\theta;\qquad \text{Cylindrical: } dV = r\,dr\,d\theta\,dz;

10

\quad \text{Spherical: } dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta,

with $x=\rho\sin\phi\cos\theta,\ y=\rho\sin\phi\sin\theta,\ z=\rho\cos\phi$ , $\phi\in[0,\pi]$ the polar angle from the $+z$ axis.

mathematician · 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.

🧠 Intuition

Think of $\iint_D f\,dA$ as stacking infinitely thin slabs. Fix $x$ : the slice $z=f(x,y)$ over the segment of $D$ at that $x$ has area $A(x)=\int f(x,y)\,dy$ . This is a wafer of thickness $dx$ and volume $A(x)\,dx$ . Sliding $x$ from $a$ to $b$ and summing wafers gives the whole solid: $\int_a^b A(x)\,dx$ . 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 $(r,\theta)$ -space is not a square in the plane — it is a curved patch of area $r\,dr\,d\theta$ , fatter far from the origin, which is exactly why the $r$ appears.

Figure

Region

D

as type I, sliced into vertical strips for iterated integration

ExampleVolume Under a Paraboloid Over a Triangle

Given: Find $\displaystyle\iint_D (x+2y)\,dA$ where $D$ is bounded by $y=2x^2$ and $y=1+x^2$ .

Step 1 — Find limits. Set $2x^2=1+x^2\Rightarrow x^2=1\Rightarrow x=\pm1$ . For $-1\le x\le1$ , the upper curve is $y=1+x^2$ , lower is $y=2x^2$ (type I).

Step 2 — Inner integral over $y$ :

\int_{2x^2}^{1+x^2}(x+2y)\,dy = \Big[xy+y^2\Big]_{2x^2}^{1+x^2}.

Upper: $x(1+x^2)+(1+x^2)^2 = x+x^3 + 1+2x^2+x^4$ . Lower: $x(2x^2)+(2x^2)^2 = 2x^3+4x^4$ . Difference: $-3x^4 -x^3 +2x^2 +x +1$ .

Step 3 — Outer integral over $x$ :

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

Odd terms ( $-x^3, +x$ ) vanish. Even terms:

\int_{-1}^1 -3x^4\,dx = -\tfrac{6}{5},\quad \int_{-1}^1 2x^2\,dx = \tfrac{4}{3},\quad \int_{-1}^1 1\,dx = 2.

Step 4 — Sum: $-\tfrac{6}{5}+\tfrac{4}{3}+2 = \tfrac{-18+20+30}{15}=\tfrac{32}{15}$ .

\boxed{\iint_D (x+2y)\,dA = \dfrac{32}{15}\ \text{(cubic units)}}

ExampleMass of a Hemisphere via Spherical Coordinates

Given: A solid hemisphere $\rho\le R$ , $z\ge0$ , has density $\delta(x,y,z)=k\,z$ (kg/m³ per meter, $k$ constant). Find its mass.

Step 1 — Convert. $z=\rho\cos\phi$ , $dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$ . Limits: $\rho\in[0,R]$ , $\phi\in[0,\pi/2]$ , $\theta\in[0,2\pi]$ .

Step 2 — Set up:

M=\int_0^{2\pi}\!\!\int_0^{\pi/2}\!\!\int_0^R k(\rho\cos\phi)\,\rho^2\sin\phi\,d\rho\,d\phi\,d\theta.

Step 3 — Separate (integrand factors):

M = k\left(\int_0^{2\pi}d\theta\right)\!\left(\int_0^{\pi/2}\cos\phi\sin\phi\,d\phi\right)\!\left(\int_0^R \rho^3\,d\rho\right).

Step 4 — Evaluate each: $\int_0^{2\pi}d\theta = 2\pi$ ; $\int_0^{\pi/2}\cos\phi\sin\phi\,d\phi = \big[\tfrac12\sin^2\phi\big]_0^{\pi/2}=\tfrac12$ ; $\int_0^R\rho^3\,d\rho=\tfrac{R^4}{4}$ .

Step 5 — Multiply: $M = k\cdot 2\pi\cdot \tfrac12\cdot \tfrac{R^4}{4} = \dfrac{\pi k R^4}{4}$ .

\boxed{M = \dfrac{\pi k R^4}{4}\ \text{kg}}

⚠️ Common mistake

$dA \ne dx\,dy$ blindly. In polar/spherical coordinates the area/volume element carries the Jacobian: forgetting the $r$ in $r\,dr\,d\theta$ or $\rho^2\sin\phi$ 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 $\int_a^b\int_{g_1(x)}^{g_2(x)} dy\,dx$ to $dx\,dy$ 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.

CounterexampleFubini Fails Without Absolute Integrability

Let $f(x,y)=\dfrac{x^2-y^2}{(x^2+y^2)^2}$ on $(0,1]\times(0,1]$ . Note $f = -\partial_y\!\left(\dfrac{y}{x^2+y^2}\right)$ , so

\int_0^1\!\int_0^1 f\,dy\,dx = \int_0^1 \frac{1}{1+x^2}\,dx = \frac{\pi}{4}.

By the antisymmetry $f(x,y)=-f(y,x)$ , reversing the order gives

\int_0^1\!\int_0^1 f\,dx\,dy = -\frac{\pi}{4}.

The two iterated integrals are both finite but unequal ( $\pi/4 \ne -\pi/4$ ). This is allowed because $\iint |f|\,dA = +\infty$ , so the Fubini hypothesis fails. The double integral itself does not exist.

connection

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 $\nabla\!\cdot\!\mathbf F$ 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 $dx\wedge dy$ encodes orientation and the change-of-variables Jacobian appears automatically. In probability, the joint-density integral is the gateway to Marginal and Conditional Distributions.

🔬 Where this appears

⏳ History

- 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

\mathbb 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.

open-problem

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 $\int f\,e^{i\lambda\phi}\,dV$ , sharp decay rates as $\lambda\to\infty$ 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.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also