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Fundamental theorem of calculus

The Fundamental Theorem of Calculus (FTC) is a foundational result in mathematical analysis that links the processes of differentiation and integration as inverse operations, enabling the computation of definite integrals through antiderivatives and revealing the rate of change of accumulated quantities.^[1] It comprises two main parts: the first part asserts that if f is a continuous function on an interval [a, b], then the function F(x) = \int_a^x f(t) \, dt is differentiable and satisfies F'(x) = f(x); the second part states that if F is any antiderivative of f (i.e., F'(x) = f(x)), then \int_a^b f(x) \, dx = F(b) - F(a).^[2] This theorem transforms the evaluation of integrals from limit-based approximations to algebraic manipulations, underpinning much of applied mathematics.^[3] The historical development of the FTC traces back to ancient and medieval scholars who intuitively grasped aspects of the integral-differential relationship, long before its formalization. For instance, Archimedes (c. 287–212 BCE) used methods of exhaustion to compute areas and volumes, laying groundwork for integration, while 14th-century scholars, including those at Merton College, Oxford, and Nicole Oresme (c. 1323–1382), employed graphical representations to connect velocity curves to distance traveled, effectively stating an early form of the theorem: the area under a velocity function equals the change in position.^[4] Earlier versions were discovered by James Gregory in 1668 and Isaac Barrow around 1670, and the theorem was independently developed and rigorously proven by Isaac Newton and Gottfried Wilhelm Leibniz during their invention of calculus in the late 17th century, with Leibniz explicitly formulating it in a 1693 paper as the general problem of quadrature.^[5]^[4] These contributions resolved longstanding puzzles in motion and change, building on precursors like the work of Bonaventura Cavalieri and Pierre de Fermat in the 1630s.^[4] In its modern form, the FTC is essential for solving differential equations, modeling physical systems such as motion and fluid flow, and advancing fields like physics, engineering, and economics, where it quantifies accumulated rates of change—such as total distance from velocity or total charge from current.^[6] The theorem's proof relies on the continuity of f and the mean value theorem, ensuring the antiderivative's existence and the integral's computability, and it extends to more general settings like Riemann-Stieltjes integrals in advanced analysis.^[3] By bridging indefinite and definite integration, the FTC not only simplifies calculations but also provides profound insights into the nature of continuous change.^[1]

Background and Intuition

Historical Development

The roots of the Fundamental Theorem of Calculus trace back to ancient precursors, notably Archimedes' method of exhaustion in the 3rd century BCE, which approximated areas under curves through successive refinements, laying groundwork for integration concepts.^[7] In the 17th century, Bonaventura Cavalieri advanced this with his method of indivisibles around the 1630s, treating areas as sums of infinitely thin lines to compute integrals like that of x^n.^[7] A pivotal early formulation emerged in 1668 when Scottish mathematician James Gregory published a geometric version of the theorem in Geometriae Pars Universalis, establishing the inverse relationship between differentiation and integration.^[8] Isaac Newton independently discovered the theorem around 1665–1666 during his development of fluxions, linking the fluxion (derivative) of an area under a curve to the ordinate (function value) at that point.^[7] Concurrently, Gottfried Wilhelm Leibniz developed the theorem independently in the 1670s while in Paris, introducing notations for differentials (dx) and integrals (\int) by 1675 to represent summation processes.^[7] Newton's work remained unpublished for decades, appearing first in De Analysi per Aequationes Numero Terminorum Infinitas (written 1669, published 1711) and later in Methodus Fluxionum et Serierum Infinitarum (written 1671, published 1736).^[7] Leibniz published his integral calculus results earlier, in a 1684 paper in Acta Eruditorum and a 1686 follow-up, sparking the calculus priority dispute.^[7] In the 19th century, Augustin-Louis Cauchy provided rigorous foundations for the theorem through his limit-based definition of the integral in works like Cours d'Analyse (1821), emphasizing continuity for differentiability.^[9] Bernhard Riemann further refined it in the 1850s with his integral definition via sums, addressing discontinuities and extending applicability to a broader class of functions.^[10]

Geometric Interpretation

The geometric interpretation of the fundamental theorem of calculus visualizes the definite integral as the net accumulated area under a continuous function curve, where the theorem connects this area to the instantaneous rate of change represented by the function's value. For a continuous function f(t) on an interval [a, x], the integral \int_a^x f(t) \, dt can be approximated by partitioning the interval into thin rectangles, each with width \Delta t and height f(t_i) at sample points, summing these rectangular areas to estimate the net region bounded by the curve, the x-axis, and the lines at t = a and t = x. As the partition refines with narrower rectangles, this sum converges to the exact net area, positive above the axis and negative below, illustrating the integral's role in measuring signed accumulation.^[11]^[12] In a typical diagram, the curve y = f(t) is plotted from a to x, with vertical strips or rectangles accumulating to form the region defining F(x) = \int_a^x f(t) \, dt, the antiderivative function; at the point x, a tangent line to the curve y = F(x) has a slope equal to the height f(x), geometrically showing how the rate of change of the accumulated area matches the original function's value at that point. This setup highlights the theorem's first part, where the derivative of the area function recovers the integrand, akin to the slope of the accumulated region's boundary aligning with the curve's ordinate.^[12] Gottfried Wilhelm Leibniz provided what is often interpreted as an early geometric demonstration in 1693, though scholarly debate exists on whether it constitutes a full proof of the theorem or a method for specific quadratures assuming the result; he conceptualized the area under a curve as a summation of infinitesimal rectangular strips, whose heights correspond to the curve's values, and linked this to tangent lines through a "law of tangency" where the slope of the area curve inversely relates to the original curve's ordinates in his geometric construction. In his diagram, inspired by Isaac Barrow's earlier work, Leibniz depicted a curve and its associated area boundary, using characteristic triangles to demonstrate how infinitesimal area increments yield tangents that reverse the integration process.^[13]^[14] The theorem's second part is illustrated geometrically as the definite integral \int_a^b f(t) \, dt equaling the vertical distance between the antiderivative values F(b) and F(a), representing the total net area between the limits as the difference in "heights" of the accumulated area function at those endpoints. This visualization shows the integral from a to b as the net region enclosed by the curve, axis, and vertical lines at a and b, directly computable via the antiderivative's evaluation without explicit summation.^[11]

Intuitive Explanation

The fundamental theorem of calculus reveals the deep connection between two core operations in calculus: integration, which represents the total accumulation of a quantity over time or space, and differentiation, which captures the instantaneous rate of change. Consider the analogy of motion: if a function f(t) describes velocity at time t, then integrating f(t) accumulates these velocities to yield the total distance traveled, much like summing up small increments of speed over intervals to find overall displacement. Conversely, differentiating the position function gives back the velocity, measuring how quickly the position is changing at any instant. This interplay shows how accumulation builds a cumulative effect, while the rate extracts the momentary contribution driving that buildup.^[15] These operations are inverses because applying one undoes the other, reversing the process of accumulation and dissection. Building up the total area under a curve through integration is akin to stacking layers to form a complete shape; then, taking the derivative reveals the slope—or height—of that shape at any point, peeling back to the original varying rate. In this way, the theorem bridges the global picture of summed changes with the local view of immediate rates, ensuring that the derivative of an accumulated quantity returns the original function.^[16] A simple illustration arises with a constant function f(t) = c, where c is a fixed value. Here, integration accumulates a steady rate, forming a rectangular region whose area grows linearly with the interval length; differentiating this accumulation then recovers the constant height c, confirming the reversal without any variation in the rate. The theorem's two parts complement each other: the first establishes that the integral from a fixed point defines an antiderivative whose derivative is the original function, while the second shows how any antiderivative can evaluate the total integral between limits by differencing values at the endpoints. This duality underscores the theorem's power in linking indefinite and definite forms of integration.^[16]^[17]

Formal Statements

First Fundamental Theorem

The first fundamental theorem of calculus establishes a profound connection between differentiation and integration by showing that the definite integral can serve as an antiderivative. Specifically, consider a function f that is continuous on the closed interval [a, b], where a < b. Define the function F by

F(x) = \int_a^x f(t) \, dt

for all x \in [a, b], using the standard notation for the definite integral with a fixed lower limit a and variable upper limit x. Under these conditions, F is continuous on [a, b] and differentiable on the open interval (a, b), with derivative F'(x) = f(x) for every x \in (a, b).^[18] The assumption of continuity of f on [a, b] is essential for ensuring that F is differentiable everywhere on (a, b) and that the derivative equals the integrand f at every point in that interval. Continuity guarantees that the accumulation of the area under f from a to x varies smoothly enough to recover f exactly upon differentiation. Without continuity, F may still be well-defined, but the equality F'(x) = f(x) might fail at points of discontinuity.^[19] If f is merely Riemann integrable on [a, b]—meaning f is bounded and continuous almost everywhere (i.e., discontinuous only on a set of Lebesgue measure zero)—then F(x) = \int_a^x f(t) \, dt remains continuous on [a, b]. However, F is differentiable (and F'(x) = f(x)) only at points where f is continuous; at discontinuities, the derivative may not exist or may not equal f(x). This weaker condition highlights that the theorem's full differentiability requires the stronger hypothesis of continuity, though Riemann integrability suffices for the existence and continuity of F.^[20] This theorem motivates viewing the definite integral as a constructive way to obtain an antiderivative of f, interpreting F(x) as the net accumulation of f from the fixed point a to x, which intuitively reverses the process of differentiation.^[21]

Second Fundamental Theorem

The second fundamental theorem of calculus provides a method for evaluating definite integrals using antiderivatives. Specifically, if f is Riemann integrable on the closed interval [a, b] and F is any antiderivative of f on (a, b) such that F'(x) = f(x) for all x \in (a, b), then

\int_a^b f(x) \, dx = F(b) - F(a).

^[12] This result holds under the assumption that f is Riemann integrable on [a, b], ensuring the definite integral exists; if f is continuous on [a, b], then any such antiderivative F is continuously differentiable on [a, b].^[12]^[22] The notation \int f(x) \, dx for the indefinite integral represents the family of all antiderivatives of f, differing only by a constant.^[23] Geometrically, this theorem interprets the definite integral \int_a^b f(x) \, dx as the total net signed area between the graph of f and the x-axis from a to b, which equals the difference in the "height" of the antiderivative F at the endpoints b and a.^[12] A key consequence of the fundamental theorem of calculus, often referred to as a related corollary, arises when the antiderivative is explicitly defined using the definite integral itself. Specifically, if f is continuous on the closed interval [a, b], then the function F(x) = \int_a^x f(t) \, dt serves as an antiderivative of f, satisfying F'(x) = f(x) for all x \in (a, b), and the one-sided derivatives at the endpoints satisfy F'_+(a) = f(a) and F'_-(b) = f(b). This allows the definite integral to be evaluated directly as \int_a^b f(x) \, dx = F(b) - F(a).^[24]^[12] The continuity assumption on f is essential, as it guarantees the differentiability of F on (a, b) and the existence of the one-sided derivatives at the endpoints equal to f(a) and f(b). Without continuity, while F remains continuous, it may not be differentiable where f fails to be continuous, preventing the equality F'(x) = f(x) at those points. For instance, if f has a jump discontinuity, such as the step function f(x) = 0 for x < 0 and f(x) = 1 for x \geq 0 on [-1, 1], then F(x) = \int_{-1}^x f(t) \, dt exhibits a corner at x = 0, where the left- and right-hand derivatives differ (F'(0^-) = 0 and F'(0^+) = 1), so F is not differentiable at the discontinuity and thus not an antiderivative of f there, even though the integral exists.^[25]^[26] This corollary facilitates direct evaluation of definite integrals by leveraging the integral-defined antiderivative, bypassing the need to compute a separate primitive function explicitly, and underscores the tight linkage between differentiation and integration under continuity.^[27]

Proofs

Proof of the First Fundamental Theorem

Assume that the function f is continuous on the closed interval [a, b]. Define the function F: [a, b] \to \mathbb{R} by

F(x) = \int_a^x f(t) \, dt

for all x \in [a, b].^[28] Since f is continuous on the compact interval [a, b], it is bounded and integrable over any subinterval, which implies that F is well-defined. Moreover, F is continuous on [a, b] because, for any x, y \in [a, b] with x < y,

|F(y) - F(x)| = \left| \int_x^y f(t) \, dt \right| \leq \int_x^y |f(t)| \, dt \leq M (y - x),

where M = \sup_{t \in [a, b]} |f(t)|, and thus \lim_{y \to x} F(y) = F(x). A similar argument holds for y < x.^[28] To establish differentiability, fix x \in (a, b) and consider the difference quotient

\frac{F(x + h) - F(x)}{h}

for h \neq 0 such that x + h \in [a, b]. First, suppose h > 0. Then,

F(x + h) - F(x) = \int_a^{x+h} f(t) \, dt - \int_a^x f(t) \, dt = \int_x^{x+h} f(t) \, dt,

\frac{F(x + h) - F(x)}{h} = \frac{1}{h} \int_x^{x+h} f(t) \, dt.

By the mean value theorem for integrals, since f is continuous on the compact interval [x, x+h], there exists some c \in (x, x+h) such that

\int_x^{x+h} f(t) \, dt = f(c) \cdot h.

Thus,

\frac{F(x + h) - F(x)}{h} = f(c).

Taking the limit as h \to 0^+, we have c \to x and, by continuity of f,

\lim_{h \to 0^+} \frac{F(x + h) - F(x)}{h} = f(x).[37]

Now suppose h < 0. Let k = -h > 0, so x + h = x - k. Then,

F(x + h) - F(x) = \int_a^{x - k} f(t) \, dt - \int_a^x f(t) \, dt = -\int_{x - k}^x f(t) \, dt,

and

\frac{F(x + h) - F(x)}{h} = \frac{ -\int_{x - k}^x f(t) \, dt }{-k} = \frac{1}{k} \int_{x - k}^x f(t) \, dt.

By the mean value theorem for integrals applied to the interval [x - k, x], there exists some d \in (x - k, x) such that

\int_{x - k}^x f(t) \, dt = f(d) \cdot k.

Thus,

\frac{F(x + h) - F(x)}{h} = f(d).

Taking the limit as h \to 0^- (equivalently, k \to 0^+), we have d \to x and, by continuity,

\lim_{h \to 0^-} \frac{F(x + h) - F(x)}{h} = f(x).[37]

Therefore, the right- and left-hand limits agree, so F is differentiable at x with

F'(x) = f(x).

^[28]

Proof of the Second Fundamental Theorem

Consider a function f that is integrable on the closed interval [a, b]. Let F be an antiderivative of f on [a, b], meaning F is differentiable on [a, b] and F'(x) = f(x) for all x \in [a, b] where f is continuous, or more generally, F'(x) = f(x) almost everywhere with respect to Lebesgue measure if f is discontinuous on a set of measure zero.^[29]^[30] To prove that \int_a^b f(t) \, dt = F(b) - F(a), construct the function G(x) = \int_a^x f(t) \, dt for x \in [a, b]. By the first fundamental theorem of calculus, G is differentiable on [a, b] and G'(x) = f(x) wherever f is continuous, or almost everywhere in the general case.^[29] Since both F and G are antiderivatives of f, their difference H(x) = F(x) - G(x) satisfies H'(x) = F'(x) - G'(x) = f(x) - f(x) = 0 almost everywhere on [a, b]. Thus, H is constant on [a, b], so H(x) = C for some constant C.^[29] To determine C, evaluate at the lower limit: G(a) = \int_a^a f(t) \, dt = 0, so H(a) = F(a) - G(a) = F(a) = C. Therefore, F(x) = G(x) + F(a) for all x \in [a, b]. Substituting x = b yields

\int_a^b f(t) \, dt = G(b) = F(b) - F(a).

If f is discontinuous but integrable (e.g., Riemann or Lebesgue integrable), the antiderivative F is absolutely continuous on [a, b], ensuring the equality holds without requiring continuity of f everywhere. Absolute continuity of F guarantees that F(b) - F(a) = \int_a^b F'(x) \, dx = \int_a^b f(x) \, dx, even at points of discontinuity of f.^[30] The related corollary combines the first and second fundamental theorems of calculus to provide a direct method for evaluating definite integrals using the indefinite integral as the antiderivative. Specifically, if f is continuous on the closed interval [a, b], define F(x) = \int_a^x f(t) \, dt. Then,

\int_a^b f(x) \, dx = F(b) - F(a).

^[28] To establish this, apply the first fundamental theorem, which guarantees that F is differentiable on [a, b] with F'(x) = f(x) for all x \in [a, b], making F an antiderivative of the continuous function f.^[28] Next, invoke the second fundamental theorem, which states that if G is any antiderivative of a continuous function f on [a, b], then \int_a^b f(x) \, dx = G(b) - G(a). Substituting G = F yields the desired equality.^[28] This approach demonstrates that constructing a separate explicit antiderivative is unnecessary; the continuity of f ensures the integral function F fulfills the role directly, simplifying computations. The dummy variable in the integral is arbitrary, so the result can be rewritten in the equivalent tautological form

\int_a^b f(x) \, dx = \int_a^b f(t) \, dt,

emphasizing notational invariance while relying on the antiderivative property of F.^[28] If f is discontinuous at some interior point of [a, b], the corollary does not apply in its classical form, as F(x) = \int_a^x f(t) \, dt is then differentiable almost everywhere (with respect to Lebesgue measure) and satisfies F'(x) = f(x) almost everywhere, but F' may fail to equal f at points of discontinuity.^[31] In such scenarios, while the definite integral exists for Riemann-integrable f (e.g., bounded with finitely many discontinuities), the direct evaluation formula using a pointwise antiderivative breaks down, requiring more advanced tools like Lebesgue integration for the full recovery of \int_a^b f(x) \, dx = F(b) - F(a).^[31]

Applications and Examples

Computing Definite Integrals

The second fundamental theorem of calculus provides a method for evaluating definite integrals of continuous functions by using antiderivatives.^[32] This approach is particularly effective for elementary functions, where antiderivatives can be found explicitly.^[33] To compute a definite integral \int_a^b f(x) \, dx using this theorem, assuming f is continuous on the closed interval [a, b], follow these steps: first, determine an antiderivative F(x) such that F'(x) = f(x); then, evaluate F(b) - F(a).^[32] The constant of integration is irrelevant, as it cancels out in the subtraction.^[33] This process transforms the integral into a straightforward algebraic evaluation at the limits. For example, consider \int_0^1 x^2 \, dx. The antiderivative is F(x) = \frac{x^3}{3}. Substituting the limits gives:

F(1) - F(0) = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3}.

^[33] Another illustration is \int_{-\pi}^{\pi} \sin x \, dx. The antiderivative is F(x) = -\cos x. Evaluating at the limits yields:

F(\pi) - F(-\pi) = [-\cos \pi] - [-\cos (-\pi)] = [-(-1)] - [-(-1)] = 1 - 1 = 0.

^[32] This result holds because \sin x is continuous on [-\pi, \pi] and the function's odd symmetry contributes to the cancellation over the symmetric interval.^[33] Such computations are routine for polynomials, trigonometric functions, and other elementary forms where antiderivatives are readily available.

Applying the First Theorem

The first fundamental theorem of calculus provides a direct method for computing the derivative of a function defined as an integral with a variable upper limit, where the integrand is continuous. Specifically, if f is continuous on an interval containing a and x, then for F(x) = \int_a^x f(t) \, dt, it follows that F'(x) = f(x).^[34] A classic example arises with the Fresnel integral, which appears in optics and engineering applications such as diffraction patterns and highway curve design. Consider the function S(x) = \int_0^x \sin\left( \frac{\pi t^2}{2} \right) dt. By the first fundamental theorem, since the integrand \sin\left( \frac{\pi t^2}{2} \right) is continuous, the derivative is S'(x) = \sin\left( \frac{\pi x^2}{2} \right).^[35] This allows straightforward differentiation of non-elementary integrals like the Fresnel function without finding an explicit antiderivative. For integrals with a variable upper limit g(x) and fixed lower limit a, the first theorem combines with the chain rule to yield the formula \frac{d}{dx} \int_a^{g(x)} f(t) \, dt = f(g(x)) g'(x), assuming f is continuous and g is differentiable.^[34] This result is a special case of the Leibniz rule for differentiation of integrals with variable limits. For instance, if F(x) = \int_1^{x^2} \sin(t) \, dt, then F'(x) = \sin(x^2) \cdot 2x.^[34] When both limits are functions of x, say h(x) (lower) and k(x) (upper), the derivative can be found by splitting the integral: \int_{h(x)}^{k(x)} f(t) \, dt = \int_a^{k(x)} f(t) \, dt - \int_a^{h(x)} f(t) \, dt, and applying the previous formula to each part. This gives \frac{d}{dx} \int_{h(x)}^{k(x)} f(t) \, dt = f(k(x)) k'(x) - f(h(x)) h'(x), provided f is continuous and the limits are differentiable.^[34] An example is F(x) = \int_x^{2x} \sin(t) \, dt, where F'(x) = \sin(2x) \cdot 2 - \sin(x) \cdot 1 = 2\sin(2x) - \sin(x).^[34]

Case Where the Corollary Fails

Consider the function f(x) = 0 if x < 0 and f(x) = 1 if x \geq 0, defined on the interval [-1, 1]. This is the Heaviside step function, which has a jump discontinuity at x = 0. The definite integral \int_{-1}^{1} f(x) \, dx = 1, as the contribution from [-1, 0) is zero and from [0, 1] is one.^[36] Define F(x) = \int_{0}^{x} f(t) \, dt. Then F(x) = 0 for x < 0 and F(x) = x for x \geq 0. The function F is continuous everywhere but not differentiable at x = 0, where the left-hand derivative is 0 and the right-hand derivative is 1. Thus, the related corollary to the first fundamental theorem—which requires f to be continuous for F'(x) = f(x) at every point in the interval—fails here due to the discontinuity of f.^[36] Nevertheless, the second fundamental theorem still applies using a different antiderivative. Let G(x) = \max(x, 0), which is continuous on [-1, 1] and satisfies G'(x) = f(x) for all x \neq 0. Then G(1) - G(-1) = 1 - 0 = 1, matching the value of the integral. This illustrates that Riemann integrability is preserved for bounded functions with finitely many discontinuities, but the choice of antiderivative affects differentiability at those points.^[36]

Theoretical Illustration

The first fundamental theorem of calculus establishes the existence of antiderivatives for continuous functions. Specifically, if f is continuous on the closed interval [a, b], then the function F(x) = \int_a^x f(t) \, dt is an antiderivative of f on (a, b), meaning F'(x) = f(x) for all x \in (a, b).^[37] This construction guarantees that every continuous function on a compact interval possesses at least one antiderivative, which is itself continuous on [a, b].^[24] A key non-computational consequence is the mean value theorem for integrals, which asserts that if f is continuous on [a, b], then there exists some c \in (a, b) such that \int_a^b f(t) \, dt = f(c)(b - a).^[38] This result follows by applying Rolle's theorem to the auxiliary function G(x) = \int_a^x f(t) \, dt - f(a)(x - a), adjusted appropriately to ensure G(b) = 0, or more directly via the mean value theorem applied to the antiderivative F(x) = \int_a^x f(t) \, dt, yielding F(b) - F(a) = F'(c)(b - a) for some c \in (a, b).^[39] The theorem highlights the theorem's role in linking average values to pointwise evaluations without explicit computation. Further implications arise in the properties of the antiderivative F. Since f is continuous on the compact interval [a, b], it is bounded, say |f(t)| \leq M for all t \in [a, b], implying that |F(y) - F(x)| \leq M|y - x| for x, y \in [a, b]. This Lipschitz condition ensures F is uniformly continuous on [a, b].^[40] Moreover, F is absolutely continuous on [a, b] because it equals the indefinite integral of an integrable function, and every absolutely continuous function is of bounded variation on [a, b].^[41] These properties underscore the theorem's contribution to understanding the regularity and structural features of functions defined via integration, emphasizing existence over explicit forms.

Extensions

Multivariable Generalizations

The fundamental theorem of calculus extends to functions of several variables through iterated integrals, allowing differentiation under the integral sign. Specifically, for a function f continuous on the rectangle [a, b] \times [c, d] and \partial_x f existing and continuous on this domain, define g(x) = \int_c^d f(x, y) \, dy. Then, g'(x) = \int_c^d \partial_x f(x, y) \, dy.^[42] This result, often called Leibniz's integral rule in the multivariable setting, mirrors the one-variable case by relating the derivative of an integral to the integral of a derivative, provided the necessary continuity conditions hold to justify interchanging the operations.^[42] A key prerequisite for working with double integrals is Fubini's theorem, which permits interchanging the order of integration under suitable conditions. For f continuous on [a, b] \times [c, d],

\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.

This equality enables the evaluation of double integrals via iterated single integrals, leveraging the fundamental theorem of calculus at each step.^[42] The theorem holds because continuity ensures the integrals exist as Riemann integrals, and the mixed partial derivatives align via Clairaut's theorem.^[42] In applications, these ideas underpin vector calculus theorems that generalize the fundamental theorem to higher dimensions. For instance, Green's theorem in the plane relates a line integral around a closed curve C bounding a region D to a double integral over D: for a vector field \mathbf{F} = (M, N) with continuous partial derivatives,

\oint_C M \, dx + N \, dy = \iint_D \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dA.

This equates the circulation of \mathbf{F} along the boundary to the flux of its curl through the region, serving as a two-dimensional analog of the theorem.^[43] For higher dimensions, Stokes' theorem provides the natural extension, linking the line integral of a vector field around a closed curve C to the surface integral of its curl over any surface S bounded by C:

\int_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}.

This generalizes the relationship between boundary integrals and interior derivatives to oriented manifolds in \mathbb{R}^3, with the divergence theorem further extending it to volumes.^[44]

Measure-Theoretic Versions

The measure-theoretic version of the fundamental theorem of calculus generalizes the classical results to the framework of Lebesgue integration on \mathbb{R}, where functions may be unbounded or discontinuous on sets of positive measure, provided they are integrable with respect to Lebesgue measure m. If f \in L^1([a,b]), the function F(x) = \int_{[a,x]} f \, dm is absolutely continuous on [a,b], differentiable Lebesgue-almost everywhere with F'(x) = f(x) almost everywhere, and satisfies F(b) - F(a) = \int_{[a,b]} f \, dm.^[45] Conversely, if F is absolutely continuous on [a,b], then F' exists almost everywhere, belongs to L^1([a,b]), and F(x) = F(a) + \int_{[a,x]} F' \, dm for all x \in [a,b].^[45] This formulation relies on the Lebesgue differentiation theorem, which asserts that for any locally integrable function f on \mathbb{R}^n, the average value over balls centered at x converges to f(x) for almost every x with respect to Lebesgue measure:

\lim_{r \to 0^+} \frac{1}{m(B(x,r))} \int_{B(x,r)} f \, dm = f(x)

almost everywhere.^[46] Applying this to the indefinite integral yields the pointwise recovery of f as the derivative almost everywhere, resolving cases where the classical Riemann FTC fails due to discontinuities.^[46] The Radon-Nikodym theorem underpins these results by providing the derivative of measures: if a signed measure \nu is absolutely continuous with respect to Lebesgue measure m on [a,b] (i.e., \nu \ll m), then there exists a unique f \in L^1([a,b]) such that d\nu = f \, dm, serving as the antiderivative in the measure-theoretic sense.^[45] For the Lebesgue-Stieltjes integral \int f \, dg, where g is of bounded variation and generates a measure \mu_g, the FTC holds if F is g-absolutely continuous: F'_{g} exists \mu_g-almost everywhere, lies in L^1_g([a,b]), and F(t) = F(a) + \int_{[a,t)} F'_{g} \, d\mu_g for almost every t \in [a,b].^[47] This extends the classical integral by allowing g to induce singular measures, with equality \int_{[a,b]} f \, dg = F(b) - F(a) when F' = f \cdot g' almost everywhere with respect to the total variation measure |\mu_g|.^[47] These measure-theoretic versions offer advantages over the Riemann FTC by accommodating integrable functions that are unbounded near endpoints or discontinuous on sets of measure zero, ensuring the theorem applies without requiring uniform continuity of the antiderivative.^[45]

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