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Iterated integral

In multivariable calculus, an iterated integral is a computational method for evaluating multiple integrals by performing a series of successive single-variable integrations, treating all other variables as constants during each step.^[1] This approach reduces the complexity of integrating functions over multi-dimensional regions, such as rectangles in the plane or rectangular boxes in higher dimensions, by breaking down the process into manageable one-dimensional integrals.^[2] For a continuous function f(x, y) over a rectangular region R = [a, b] \times [c, d], the double integral \iint_R f(x, y) \, dA can be expressed as the iterated integral \int_a^b \left( \int_c^d f(x, y) \, dy \right) dx, where the inner integral is evaluated first with respect to y (holding x fixed), and the outer integral follows with respect to x.^[3] The order of integration can be reversed to \int_c^d \left( \int_a^b f(x, y) \, dx \right) dy, yielding the same result under appropriate conditions.^[4] Fubini's theorem provides the theoretical foundation for this equivalence, stating that if f(x, y) is continuous on the closed rectangle R, then \iint_R f(x, y) \, dA = \int_a^b \int_c^d f(x, y) \, dy \, dx = \int_c^d \int_a^b f(x, y) \, dx \, dy.^[2] This theorem, which extends to higher dimensions for continuous functions over rectangular domains, ensures that the multiple integral equals the iterated integral regardless of the integration order, provided the function satisfies the continuity requirement.^[5] Iterated integrals are essential for practical applications in mathematics and related fields, including the computation of areas of plane regions, volumes of solids bounded by surfaces, and average values of functions over domains.^[1] For instance, the area of a rectangle [0, 2] \times [0, 3] is given by the iterated integral \int_0^2 \int_0^3 1 \, dy \, dx = 6, demonstrating how the method quantifies geometric properties through integration.^[4] In higher dimensions, for example, double integrals for volumes under a surface z = f(x, y) over a region in the xy-plane, or triple integrals for volumes of solids in three dimensions, the process iterates similarly, integrating first with respect to one variable, then another, and finally the third.^[3]

Fundamentals

Definition

An iterated integral is a method for evaluating a multiple integral by successively integrating a multivariable function with respect to one variable at a time, while treating the remaining variables as constants.^[6] This approach reduces the computation of integrals over multidimensional domains to a sequence of single-variable integrals.^[7] For a double iterated integral in the Riemann sense, consider a function f(x,y) defined on a region where, for x \in [a,b], y ranges from g(x) to h(x). The formal definition is given by

\int_{a}^{b} \left( \int_{g(x)}^{h(x)} f(x,y) \, dy \right) dx,

where the inner integral is performed with respect to y for each fixed x, yielding a function of x, which is then integrated with respect to x in the outer integral.^[7] This construction builds directly on the Riemann integral for single variables but extends it to functions of two or more variables.^[6] In the Lebesgue framework, the iterated integral is defined analogously using Lebesgue measures on product spaces, allowing integration over \mathbb{R}^n by iterating one-dimensional Lebesgue integrals, such as \int_{\mathbb{R}} \left( \int_{\mathbb{R}} f(x,y) \, dy \right) dx for nonnegative measurable functions.^[8] This assumes familiarity with definite integrals in one variable, whether Riemann or Lebesgue.^[9]

Notation

The standard notation for a double integral of a function f(x,y) over a region D in the plane is \iint_D f(x,y) \, dA, where dA denotes the differential area element. This is typically expressed in iterated form as \int_a^b \int_{g(x)}^{h(x)} f(x,y) \, dy \, dx, with the inner integral taken with respect to y over bounds that may depend on x, such as g(x) \leq y \leq h(x), and the outer integral with respect to x from constant limits a to b.^[10] The order of integration can be reversed, yielding \int_c^d \int_{p(y)}^{q(y)} f(x,y) \, dx \, dy, where the inner bounds now depend on y.^[2] In this notation, the differential following each integral sign indicates the variable of integration, with dy \, dx signifying integration first in y and then in x. For rectangular regions, where D = [a,b] \times [c,d], the bounds are constant, simplifying the iterated integral to \int_a^b \int_c^d f(x,y) \, dy \, dx or the reversed order \int_c^d \int_a^b f(x,y) \, dx \, dy.^[11] Non-rectangular domains use variable-dependent bounds to describe the region's geometry, such as vertical strips defined by functions g(x) and h(x) or horizontal strips via p(y) and q(y).^[10] The notation extends naturally to triple integrals over a region E in three-dimensional space, denoted \iiint_E f(x,y,z) \, dV, where dV is the volume element. An iterated form is \int_a^b \int_{c(x)}^{d(x)} \int_{e(x,y)}^{f(x,y)} f(x,y,z) \, dz \, dy \, dx, with nested bounds reflecting the region's description, such as z between surfaces e(x,y) and f(x,y), y between curves c(x) and d(x), and x from a to b.^[11] For rectangular boxes [a_1,b_1] \times [a_2,b_2] \times [a_3,b_3], constant bounds apply similarly.^[10] The conventions for these notations, including the use of dependent bounds for non-rectangular domains, facilitate the representation of integrals over complex geometries while maintaining the sequential nature of iteration. The evolution of this notation occurred in the 19th century, building on the foundational work of Augustin-Louis Cauchy, who addressed double integrals in his studies of definite integrals around 1823, and Bernhard Riemann, who advanced the rigorous theory of integration in 1854.^[12]^[13]

Theoretical Foundations

Fubini's Theorem

Fubini's theorem establishes the equivalence between multiple integrals and iterated integrals under suitable conditions, serving as a foundational result in multivariable calculus and measure theory. In the context of Riemann integrals, consider a function f(x,y) that is continuous on a closed and bounded region D in \mathbb{R}^2. Then the double integral over D equals the corresponding iterated integrals in either order:

\iint_D f(x,y) \, dA = \int_a^b \int_{g(x)}^{h(x)} f(x,y) \, dy \, dx = \int_c^d \int_{p(y)}^{q(y)} f(x,y) \, dx \, dy,

where the bounds describe the region D as a \leq x \leq b, g(x) \leq y \leq h(x) or c \leq y \leq d, p(y) \leq x \leq q(y).^[14] This result generalizes to Lebesgue integrals over product measure spaces. Let (X, \mathcal{S}, \mu) and (Y, \mathcal{T}, \nu) be \sigma-finite measure spaces, and let f: X \times Y \to \mathbb{R} be measurable with \int_{X \times Y} |f| \, d(\mu \times \nu) < \infty. Then f is integrable over X \times Y, the sections y \mapsto f(x,y) are integrable over Y for \mu-almost every x \in X, the sections x \mapsto f(x,y) are integrable over X for \nu-almost every y \in Y, and

\int_{X \times Y} f \, d(\mu \times \nu) = \int_X \left( \int_Y f(x,y) \, d\nu(y) \right) d\mu(x) = \int_Y \left( \int_X f(x,y) \, d\mu(x) \right) d\nu(y).

^[8] A proof sketch proceeds by reducing the multiple integral to one dimension through slicing the product space along coordinate planes, where the product measure on rectangles A \times B is \mu(A) \nu(B). Assuming absolute integrability, Tonelli's theorem applies to |f| to establish integrability of sections almost everywhere; the general case follows by decomposing f = f^+ - f^- and linearity of the integral.^[8] The theorem is named after the Italian mathematician Guido Fubini, who established a general version in 1907. Its development traces roots to Fubini's contributions on multiple integrals and subsequent refinements by Leonida Tonelli in 1909 for non-negative measurable functions.^[15]

Conditions for Interchange

The absolute integrability condition plays a central role in justifying the interchange of integration order for iterated integrals. Specifically, if f(x,y) is measurable and \iint_D |f(x,y)| \, dA < \infty over a region D \subseteq \mathbb{R}^2, then both iterated integrals \int \left( \int f(x,y) \, dy \right) dx and \int \left( \int f(x,y) \, dx \right) dy exist, are equal, and coincide with the double integral \iint_D f(x,y) \, dA, even when f is discontinuous on a set of positive measure.^[16] This condition, part of Fubini's theorem in its general form, ensures the validity of the interchange by controlling the potential for cancellation in the integrand that could lead to discrepancies. Without absolute integrability, the iterated integrals may exist but yield different values depending on the order, highlighting the limitations of unrestricted interchange. A classic continuous counterexample occurs with f(x,y) = \frac{xy(x^2 - y^2)}{(x^2 + y^2)^3} for (x,y) \neq (0,0) and f(0,0) = 0, integrated over the rectangle R = [0,2] \times [0,1]. Here, \int_0^1 \left( \int_0^2 f(x,y) \, dx \right) dy = -\frac{1}{20}, while \int_0^2 \left( \int_0^1 f(x,y) \, dy \right) dx = \frac{1}{5}, yet \iint_R |f(x,y)| \, dA = \infty, so the double integral is undefined.^[17] Such cases demonstrate that conditional convergence in multiple dimensions can cause order dependence, analogous to one-dimensional improper integrals. For non-negative functions, Tonelli's theorem provides a more permissive framework, allowing interchange without requiring absolute integrability. If f(x,y) \geq 0 is measurable over D, then the iterated integrals \int \left( \int f(x,y) \, dy \right) dx and \int \left( \int f(x,y) \, dx \right) dy both equal the double integral \iint_D f(x,y) \, dA (which may be infinite), and the inner integrals exist (possibly infinite) almost everywhere.^[16] This result extends Fubini's theorem by leveraging monotonicity to avoid issues with sign changes. Pathological cases often arise with discontinuous or non-absolutely integrable functions, where the iterated integrals may differ from the multiple integral or fail to agree with each other. For example, on \mathbb{N} \times \mathbb{N} with counting measure, define f(m,n) = 1 if n = m, f(m,n) = -1 if n = m+1, and $0 otherwise; then \sum_m \left( \sum_n f(m,n) \right) = 0, but \sum_n \left( \sum_m f(m,n) \right) = 1, and the double sum is undefined due to lack of absolute convergence.^[16] Similar discrepancies occur in continuous settings with non-σ-finite measures, underscoring the need for the aforementioned conditions to ensure equivalence.

Computation

Basic Evaluation

The basic evaluation of an iterated integral proceeds by first computing the inner integral, treating the variable of the outer integral as a constant, and then evaluating the resulting single integral with respect to the outer variable. This method applies directly to integrals over rectangular regions, where the limits of integration are constants independent of the other variable. For continuous functions over such regions, the result equals the double integral over the region.^[18] Consider the straightforward example of evaluating the iterated integral of the function f(x, y) = x + y over the unit square R = [0, 1] \times [0, 1]:

\int_0^1 \int_0^1 (x + y) \, dy \, dx.

The inner integral with respect to y, treating x as constant, is

\int_0^1 (x + y) \, dy = \left[ x y + \frac{y^2}{2} \right]_0^1 = x \cdot 1 + \frac{1}{2} - 0 = x + \frac{1}{2}.

The outer integral is then

\int_0^1 \left( x + \frac{1}{2} \right) \, dx = \left[ \frac{x^2}{2} + \frac{x}{2} \right]_0^1 = \frac{1}{2} + \frac{1}{2} = 1.

This yields \iint_R (x + y) \, dA = 1.^[18] Another representative example over a rectangular region involves a trigonometric function, such as evaluating \iint_R \sin x \cos y \, dA where R = [0, \pi/2] \times [0, \pi/2], set up as the iterated integral

\int_0^{\pi/2} \int_0^{\pi/2} \sin x \cos y \, dy \, dx.

The inner integral with respect to y, treating \sin x as constant, is

\int_0^{\pi/2} \cos y \, dy = \left[ \sin y \right]_0^{\pi/2} = 1 - 0 = 1,

so the integrand simplifies to \sin x \cdot 1. The outer integral is

\int_0^{\pi/2} \sin x \, dx = \left[ -\cos x \right]_0^{\pi/2} = -\cos(\pi/2) + \cos 0 = 0 + 1 = 1.

Thus, the value is 1.^[2] When setting up basic evaluations over rectangular regions, select the order of integration based on which inner integral is simpler to compute, though both orders yield the same result under suitable conditions; constants with respect to the inner variable integrate straightforwardly by pulling them outside the integral. A common error is mismatching the differential with the inner limits, such as using dx for an inner integral with y-bounds, which leads to incorrect setup and evaluation. Another frequent mistake is failing to treat the outer variable as constant in the inner integral, resulting in erroneous antiderivatives.^[2]

Order Dependence

In cases where the integrand is not absolutely integrable, the order of integration in iterated integrals can lead to different results, highlighting the importance of the conditions in Fubini's theorem.^[19] A classic example is the function f(x,y) = \frac{x - y}{(x + y)^3} over the unit square [0,1] \times [0,1]. Computing the iterated integral first with respect to y then x yields

\int_0^1 \left( \int_0^1 \frac{x - y}{(x + y)^3} \, dy \right) dx = 0,

while reversing the order gives

\int_0^1 \left( \int_0^1 \frac{x - y}{(x + y)^3} \, dx \right) dy = \frac{\pi}{4}.

This discrepancy arises because the integral is only conditionally convergent; the absolute integral \iint_{[0,1]^2} \left| \frac{x - y}{(x + y)^3} \right| \, dx \, dy diverges due to the singularity along the line x + y = 0, violating the absolute integrability condition required for interchanging the order under Fubini's theorem.^[19] In one order, the inner integral remains bounded for fixed outer variable, but in the other, it exhibits behavior leading to the divergent absolute value, emphasizing that the inner integral may diverge for certain fixed values of the outer variable when absolute convergence fails.^[20] Similar issues occur with improper integrals over unbounded domains. For instance, consider \iint_{[0,\infty)^2} e^{-xy} \, dx \, dy. The iterated integral in either order diverges: integrating first with respect to x gives \int_0^\infty \frac{1 - e^{-y \cdot \infty}}{y} \, dy = \int_0^\infty \frac{1}{y} \, dy, which diverges logarithmically, and symmetrically for the reverse order. However, the multiple integral can converge when interpreted as a limit over expanding finite regions in a symmetric manner, such as \lim_{R \to \infty} \int_0^R \int_0^R e^{-xy} \, dy \, dx, though careful analysis shows it still diverges overall without absolute convergence; this illustrates the need for caution in improper settings where one order may appear to diverge while the double integral requires principal value or other regularization to assess convergence. To avoid such order dependence, practitioners should always verify absolute integrability before interchanging the order of integration, as guaranteed by Fubini's theorem for absolutely integrable functions. For non-negative integrands, Tonelli's theorem provides a safer alternative, allowing interchange even without absolute integrability, since the integrals coincide (finite or infinite).

Extensions

Higher Dimensions

Iterated integrals extend naturally to higher dimensions, allowing the evaluation of multiple integrals over regions in \mathbb{R}^n for n \geq 3. In three dimensions, the triple iterated integral over a region E \subset \mathbb{R}^3 is denoted as \iiint_E f(x,y,z) \, dV, which can be expressed as an iterated integral such as \int_a^b \int_{c(x)}^{d(x)} \int_{e(x,y)}^{f(x,y)} f(x,y,z) \, dz \, dy \, dx, where the limits depend on the geometry of E.^[8] This notation builds on the double integral by successively integrating with respect to each variable, starting from the innermost integral. As the dimension increases, setting up the bounds for the iterated integrals becomes more complex, particularly for non-rectangular regions like spheres or cylinders, where the limits for inner variables depend nonlinearly on outer ones. To address this, alternative coordinate systems such as cylindrical coordinates (r, \theta, z) or spherical coordinates (\rho, \theta, \phi) are often employed, transforming the volume element to r \, dr \, d\theta \, dz or \rho^2 \sin \phi \, d\rho \, d\theta \, d\phi, respectively, which simplifies bounds for symmetric volumes but requires careful Jacobian adjustments.^[21] Fubini's theorem generalizes to n dimensions, stating that for a measurable function f: \mathbb{R}^n \to \mathbb{R} that is absolutely integrable (i.e., \int_{\mathbb{R}^n} |f| \, d\lambda_n < \infty, where \lambda_n is the Lebesgue measure), the multiple integral equals any iterated integral: \int_{\mathbb{R}^n} f \, d\lambda_n = \int_{\mathbb{R}} \cdots \int_{\mathbb{R}} f(x_1, \dots, x_n) \, dx_1 \cdots dx_n.^[8] This holds over product measures on \mathbb{R}^n, enabling order interchange under the absolute integrability condition. For example, the volume of the unit ball in \mathbb{R}^3, given by \iiint_{x^2 + y^2 + z^2 \leq 1} 1 \, dV, can be set up in spherical coordinates as \int_0^{2\pi} \int_0^\pi \int_0^1 \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta, illustrating the iterated structure without computing the full value here.^[21]

Applications in Analysis

Iterated integrals play a central role in Fourier analysis, particularly in computing the Fourier transform of functions over \mathbb{R}^n. For an absolutely integrable function f: \mathbb{R}^n \to \mathbb{C}, the Fourier transform \hat{f}(\xi) = \int_{\mathbb{R}^n} f(x) e^{-i \langle \xi, x \rangle} \, dx can be evaluated as an iterated integral by successive integration along each coordinate axis, leveraging Fubini's theorem under suitable conditions.^[22] This approach facilitates the decomposition of multidimensional transforms into one-dimensional integrals, which is essential for analyzing signals and solving partial differential equations in harmonic analysis.^[23] In multivariable calculus, iterated integrals are instrumental in the change of variables formula, where the Jacobian determinant adjusts the measure to simplify integration over transformed regions. For instance, converting a double integral over a disk in Cartesian coordinates to polar coordinates yields \iint_D x^2 \, dA = \int_0^{2\pi} \int_0^1 (r \cos \theta)^2 r \, dr \, d\theta, with the extra r factor arising from the absolute value of the Jacobian of the transformation x = r \cos \theta, y = r \sin \theta.^[24] This technique is widely used to evaluate integrals over non-rectangular domains, such as ellipses or sectors, by aligning the coordinates with the region's symmetry.^[25] In probability theory, iterated integrals compute expected values and marginal distributions from joint probability density functions. For jointly continuous random variables X and Y with joint density f_{X,Y}(x,y), the expected value E[X] = \iint x f_{X,Y}(x,y) \, dx \, dy is obtained by iterating the integral first over y (for fixed x) and then over x, assuming the density integrates to 1 over the plane.^[26] Similarly, the marginal density of X is f_X(x) = \int f_{X,Y}(x,y) \, dy, derived via iterated integration, which underpins computations of conditional expectations and variance in multivariate stochastic processes.^[27] Iterated integrals also appear in the study of differential equations through Green's theorem, which equates a line integral around a positively oriented, piecewise-smooth simple closed curve C to a double integral over the enclosed region D: \int_C P \, dx + Q \, dy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA.^[28] The double integral on the right is typically evaluated as an iterated integral, \int_a^b \int_{g(x)}^{h(x)} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dy \, dx, providing a powerful tool for verifying solutions to Poisson's equation and computing fluxes in vector calculus applications.^[29] In modern functional analysis, iterated integrals extend to Bochner integrals for functions valued in Banach spaces, enabling the integration of vector-valued maps where simple pointwise limits fail. The Bochner integral of a strongly measurable function f: \Omega \to X (with X a Banach space) is defined via approximation by simple functions, and iterated versions allow sequential integration over product measures, as in representations of fractional integrals in infinite-dimensional settings.^[30] This framework supports applications in operator theory and stochastic evolution equations, where Banach-valued processes require such generalized integration.^[31]

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