Fundamental theorem of calculus

Fundamental theorem of calculus Part of a series of articles about Calculus Fundamental theorem Leibniz integral rule Limits of functionsContinuity Mean value theoremRolle's theorem show Differential show Integral show Series show Vector show Multivariable show Advanced show Specialized show Miscellaneous vte The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating the gradient) with the concept of integrating a function (calculating the area under the curve). The two operations are inverses of each other apart from a constant value which is dependent on where one starts to compute area.

The first part of the theorem, sometimes called the first fundamental theorem of calculus, states that one of the antiderivatives (also known as an indefinite integral), say F, of some function f may be obtained as the integral of f with a variable bound of integration. This implies the existence of antiderivatives for continuous functions.[1] Conversely, the second part of the theorem, sometimes called the second fundamental theorem of calculus, states that the integral of a function f over some interval can be computed by using any one, say F, of its infinitely many antiderivatives. This part of the theorem has key practical applications, because explicitly finding the antiderivative of a function by symbolic integration avoids numerical integration to compute integrals.

Contents 1 History 2 Geometric meaning 3 Physical intuition 4 Formal statements 4.1 First part 4.2 Corollary 4.3 Second part 5 Proof of the first part 6 Proof of the corollary 7 Proof of the second part 8 Relationship between the parts 9 Examples 9.1 Computing a particular integral 9.2 Using the first part 9.3 An integral where the corollary is insufficient 9.4 Theoretical example 10 Generalizations 11 See also 12 Notes 13 References 13.1 Bibliography 14 Further reading 15 External links History See also: History of calculus The fundamental theorem of calculus relates differentiation and integration, showing that these two operations are essentially inverses of one another. Before the discovery of this theorem, it was not recognized that these two operations were related. Ancient Greek mathematicians knew how to compute area via infinitesimals, an operation that we would now call integration. The origins of differentiation likewise predate the fundamental theorem of calculus by hundreds of years; for example, in the fourteenth century the notions of continuity of functions and motion were studied by the Oxford Calculators and other scholars. The historical relevance of the fundamental theorem of calculus is not the ability to calculate these operations, but the realization that the two seemingly distinct operations (calculation of geometric areas, and calculation of gradients) are actually closely related.

The first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character,[2] was by James Gregory (1638–1675).[3][4] Isaac Barrow (1630–1677) proved a more generalized version of the theorem,[5] while his student Isaac Newton (1642–1727) completed the development of the surrounding mathematical theory. Gottfried Leibniz (1646–1716) systematized the knowledge into a calculus for infinitesimal quantities and introduced the notation used today.

Geometric meaning The area shaded in red stripes is close to h times f(x). Alternatively, if the function A(x) were known, this area would be exactly A(x + h) − A(x). These two values are approximately equal, particularly for small h.

For a continuous function y = f(x) whose graph is plotted as a curve, each value of x has a corresponding area function A(x), representing the area beneath the curve between 0 and x. The function A(x) may not be known, but it is given that it represents the area under the curve.

The area under the curve between x and x + h could be computed by finding the area between 0 and x + h, then subtracting the area between 0 and x. In other words, the area of this "strip" would be A(x + h) − A(x).

There is another way to estimate the area of this same strip. As shown in the accompanying figure, h is multiplied by f(x) to find the area of a rectangle that is approximately the same size as this strip. So: {displaystyle A(x+h)-A(x)approx f(x)cdot h} In fact, this estimate becomes a perfect equality if we add the red portion of the "excess" area shown in the diagram. So: {displaystyle A(x+h)-A(x)=f(x)cdot h+({text{Red Excess}})} Rearranging terms: {displaystyle f(x)={frac {A(x+h)-A(x)}{h}}-{frac {text{Red Excess}}{h}}.} As h approaches 0 in the limit, the last fraction can be shown to go to zero.[6] This is true because the area of the red portion of excess region is less than or equal to the area of the tiny black-bordered rectangle. More precisely, {displaystyle left|f(x)-{frac {A(x+h)-A(x)}{h}}right|={frac {|{text{Red Excess}}|}{h}}leq {frac {h(f(x+h_{1})-f(x+h_{2}))}{h}}=f(x+h_{1})-f(x+h_{2}),} where {displaystyle x+h_{1}} and {displaystyle x+h_{2}} are points where f reaches its maximum and its minimum, respectively, in the interval [x, x + h]. By the continuity of f, the latter expression tends to zero as h does. Therefore, the left-hand side tends to zero as h does, which implies {displaystyle f(x)=lim _{hto 0}{frac {A(x+h)-A(x)}{h}}.} This implies f(x) = A′(x). That is, the derivative of the area function A(x) exists and is the original function f(x); so, the area function is simply an antiderivative of the original function. Computing the derivative of a function and finding the area under its curve are "opposite" operations. This is the crux of the Fundamental Theorem of Calculus.

Physical intuition Intuitively, the theorem states that the sum of infinitesimal changes in a quantity over time (or over some other variable) adds up to the net change in the quantity.

Imagine, for example, using a stopwatch to mark off tiny increments of time as a car travels down a highway. Imagine also looking at the car's speedometer as it travels, so that at every moment you know the velocity of the car. To understand the power of this theorem, imagine also that you are not allowed to look out of the window of the car, so that you have no direct evidence of how far the car has traveled.

For any tiny interval of time in the car, you could calculate how far the car has traveled in that interval by multiplying the current speed of the car times the length of that tiny interval of time. (This is because distance = speed × time.) Now imagine doing this instant after instant, so that for every tiny interval of time you know how far the car has traveled. In principle, you could then calculate the total distance traveled in the car (even though you never looked out of the window) by summing-up all those tiny distances.

{displaystyle {text{distance traveled}}=sum {text{the velocity at any instant}}times {text{a tiny interval of time}}} In other words, {displaystyle {text{distance traveled}}=sum v(t)times Delta t} On the right hand side of this equation, as {displaystyle Delta t} becomes infinitesimally small, the operation of "summing up" corresponds to integration. So what we've shown is that the integral of the velocity function can be used to compute how far the car has traveled.

Now remember that the velocity function is the derivative of the position function. So what we have really shown is that integrating the velocity recovers the original position function. This is the basic idea of the theorem: that integration and differentiation are closely related operations, each essentially being the inverse of the other.

In other words, in terms of one's physical intuition, the theorem states that the sum of the changes in a quantity over time (such as position, as calculated by multiplying velocity times time) adds up to the total net change in the quantity. Or to put this more generally: Given a quantity {displaystyle x} that changes over some variable {displaystyle t} , and Given the velocity {displaystyle v(t)} with which that quantity changes over that variable then the idea that "distance equals speed times time" corresponds to the statement {displaystyle dx=v(t),dt} meaning that one can recover the original function {displaystyle x(t)} by integrating its derivative, the velocity {displaystyle v(t)} , over {displaystyle t} .

Formal statements There are two parts to the theorem. The first part deals with the derivative of an antiderivative, while the second part deals with the relationship between antiderivatives and definite integrals.

First part This part is sometimes referred to as the first fundamental theorem of calculus.[7] Let f be a continuous real-valued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], by {displaystyle F(x)=int _{a}^{x}f(t),dt.} Then F is uniformly continuous on [a, b] and differentiable on the open interval (a, b), and {displaystyle F'(x)=f(x)} for all x in (a, b) so F is an antiderivative of f.

Corollary Fundamental theorem of calculus (animation) The fundamental theorem is often employed to compute the definite integral of a function {displaystyle f} for which an antiderivative {displaystyle F} is known. Specifically, if {displaystyle f} is a real-valued continuous function on {displaystyle [a,b]} and {displaystyle F} is an antiderivative of {displaystyle f} in {displaystyle [a,b]} then {displaystyle int _{a}^{b}f(t),dt=F(b)-F(a).} The corollary assumes continuity on the whole interval. This result is strengthened slightly in the following part of the theorem.

Second part This part is sometimes referred to as the second fundamental theorem of calculus[8] or the Newton–Leibniz axiom.

Let {displaystyle f} be a real-valued function on a closed interval {displaystyle [a,b]} and {displaystyle F} a continuous function on {displaystyle [a,b]} which is an antiderivative of {displaystyle f} in {displaystyle (a,b)} : {displaystyle F'(x)=f(x).} If {displaystyle f} is Riemann integrable on {displaystyle [a,b]} then {displaystyle int _{a}^{b}f(x),dx=F(b)-F(a).} The second part is somewhat stronger than the corollary because it does not assume that {displaystyle f} is continuous.

When an antiderivative {displaystyle F} of {displaystyle f} exists, then there are infinitely many antiderivatives for {displaystyle f} , obtained by adding an arbitrary constant to {displaystyle F} . Also, by the first part of the theorem, antiderivatives of {displaystyle f} always exist when {displaystyle f} is continuous.

Proof of the first part For a given f(t), define the function F(x) as {displaystyle F(x)=int _{a}^{x}f(t),dt.} For any two numbers x1 and x1 + Δx in [a, b], we have {displaystyle F(x_{1})=int _{a}^{x_{1}}f(t),dt} and {displaystyle F(x_{1}+Delta x)=int _{a}^{x_{1}+Delta x}f(t),dt.} Subtracting the two equalities gives {displaystyle F(x_{1}+Delta x)-F(x_{1})=int _{a}^{x_{1}+Delta x}f(t),dt-int _{a}^{x_{1}}f(t),dt.}         (1) The sum of the areas of two adjacent regions is equal to the area of both regions combined, thus: {displaystyle int _{a}^{x_{1}}f(t),dt+int _{x_{1}}^{x_{1}+Delta x}f(t),dt=int _{a}^{x_{1}+Delta x}f(t),dt.} Manipulating this equation gives {displaystyle int _{a}^{x_{1}+Delta x}f(t),dt-int _{a}^{x_{1}}f(t),dt=int _{x_{1}}^{x_{1}+Delta x}f(t),dt.} Substituting the above into (1) results in {displaystyle F(x_{1}+Delta x)-F(x_{1})=int _{x_{1}}^{x_{1}+Delta x}f(t),dt.}         (2) According to the mean value theorem for integration, there exists a real number {displaystyle cin [x_{1},x_{1}+Delta x]} such that {displaystyle int _{x_{1}}^{x_{1}+Delta x}f(t),dt=f(c)cdot Delta x.} To keep the notation simple, we write just {displaystyle c} , but one should keep in mind that, for a given function {displaystyle f} , the value of {displaystyle c} depends on {displaystyle x_{1}} and on {displaystyle Delta x,} but is always confined to the interval {displaystyle [x_{1},x_{1}+Delta x]} . Substituting the above into (2) we get {displaystyle F(x_{1}+Delta x)-F(x_{1})=f(c)cdot Delta x.} Dividing both sides by {displaystyle Delta x} gives {displaystyle {frac {F(x_{1}+Delta x)-F(x_{1})}{Delta x}}=f(c).} The expression on the left side of the equation is Newton's difference quotient for F at x1.

Take the limit as {displaystyle Delta xto 0} on both sides of the equation.

{displaystyle lim _{Delta xto 0}{frac {F(x_{1}+Delta x)-F(x_{1})}{Delta x}}=lim _{Delta xto 0}f(c).} The expression on the left side of the equation is the definition of the derivative of F at x1.

{displaystyle F'(x_{1})=lim _{Delta xto 0}f(c).}         (3) To find the other limit, we use the squeeze theorem. The number c is in the interval [x1, x1 + Δx], so x1 ≤ c ≤ x1 + Δx.

Also, {textstyle lim _{Delta xto 0}x_{1}=x_{1}} and {textstyle lim _{Delta xto 0}x_{1}+Delta x=x_{1}.} Therefore, according to the squeeze theorem, {displaystyle lim _{Delta xto 0}c=x_{1}.} The function f is continuous at x1, the limit can be taken inside the function: {displaystyle lim _{Delta xto 0}f(c)=f(x_{1}).} Substituting into (3), we get {displaystyle F'(x_{1})=f(x_{1}).} which completes the proof.[9] Proof of the corollary Suppose F is an antiderivative of f, with f continuous on [a, b]. Let {displaystyle G(x)=int _{a}^{x}f(t),dt.} By the first part of the theorem, we know G is also an antiderivative of f. Since F′ − G′ = 0 the mean value theorem implies that F − G is a constant function, that is, there is a number c such that G(x) = F(x) + c for all x in [a, b]. Letting x = a, we have {displaystyle F(a)+c=G(a)=int _{a}^{a}f(t),dt=0,} which means c = −F(a). In other words, G(x) = F(x) − F(a), and so {displaystyle int _{a}^{b}f(x),dx=G(b)=F(b)-F(a).} Proof of the second part This is a limit proof by Riemann sums.

To begin, we recall the mean value theorem. Stated briefly, if F is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists some c in (a, b) such that {displaystyle F'(c)(b-a)=F(b)-F(a).} Let f be (Riemann) integrable on the interval [a, b], and let f admit an antiderivative F on (a, b) such that F is continuous on [a, b]. Begin with the quantity F(b) − F(a). Let there be numbers x1, ..., xn such that {displaystyle a=x_{0}

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