From Wikipedia, the free encyclopedia
The
fundamental theorem of calculus is a
theorem that links the concept of
differentiating a
function with the concept of
integrating a function.
The first part of the theorem, sometimes called the
first fundamental theorem of calculus, states that one of the
antiderivatives (also called
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.
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. This provides generally a better numerical accuracy.
History
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 velocities) are actually closely
related.
The first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character, was by
James Gregory (1638–1675).
Isaac Barrow (1630–1677) proved a more generalized version of the theorem, 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 can be estimated as h times f(x).
Alternatively, if the function A(x) were known, it could be
computed exactly as 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:
In fact, this estimate becomes a perfect equality if we add the red portion of the "excess" area shown in the diagram. So:
Rearranging terms:
- .
As
h approaches 0 in the
limit, the last fraction can be shown to go to zero.
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,
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
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 simply 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 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've never looked out the window) by simply summing-up all those tiny distances.
- distance traveled = the velocity at any instant a tiny interval of time
In other words,
- distance traveled =
On the right hand side of this equation, as
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 simply the derivative
of the position function. So what we have really shown is that
integrating the velocity simply 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
simply 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 that changes over some variable , and
- Given the velocity with which that quantity changes over that variable
then the idea that "distance equals speed times time" corresponds to the statement
meaning that one can recover the original function
by integrating its derivative, the velocity
, over
.
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.
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
Then,
F is uniformly continuous on [
a,
b], differentiable on the open interval
(a, b), and
for all
x in (
a,
b).
Corollary
Fundamental theorem of calculus (animation)
The fundamental theorem is often employed to compute the definite integral of a function
for which an antiderivative
is known. Specifically, if
is a real-valued continuous function on
and
is an antiderivative of
in
then
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
be a real-valued function on a
closed interval and
an antiderivative of
in
:
If
is
Riemann integrable on
then
The second part is somewhat stronger than the corollary because it does not assume that
is continuous.
When an antiderivative
exists, then there are infinitely many antiderivatives for
, obtained by adding an arbitrary constant to
. Also, by the first part of the theorem, antiderivatives of
always exist when
is continuous.
Proof of the first part
For a given
f(
t), define the function
F(
x) as
For any two numbers
x1 and
x1 + Δ
x in [
a,
b], we have
and
Subtracting the two equalities gives
It can be shown that
- (The sum of the areas of two adjacent regions is equal to the area of both regions combined.)
Manipulating this equation gives
Substituting the above into (1) results in
According to the
mean value theorem for integration, there exists a real number
such that
To keep the notation simple, we write just
, but one should keep in mind that, for a given function
, the value of
depends on
and on
but is always confined to the interval
. Substituting the above into (2) we get
Dividing both sides by
gives
- The expression on the left side of the equation is Newton's difference quotient for F at x1.
Take the limit as
→ 0 on both sides of the equation.
The expression on the left side of the equation is the definition of the derivative of
F at
x1.
To find the other limit, we use the
squeeze theorem. The number
c is in the interval [
x1,
x1 + Δ
x], so
x1 ≤
c ≤
x1 + Δ
x.
Also,
and
Therefore, according to the squeeze theorem,
Substituting into (3), we get
The function
f is continuous at
c, so the limit can be taken inside the function. Therefore, we get
which completes the proof.
Proof of the corollary
Suppose
F is an antiderivative of
f, with
f continuous on
[a, b]. Let
- .
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, i. e. there is a number
c such that
G(x) = F(x) + c, for all
x in
[a, b]. Letting
x = a, we have
which means
c = − F(a). In other words,
G(x) = F(x) − F(a), and so
Proof of the second part
This is a limit proof by
Riemann sums.
Let
f be (Riemann) integrable on the interval
[a, b], and let
f admit an antiderivative
F on
[a, b]. Begin with the quantity
F(b) − F(a). Let there be numbers
x1, ...,
xn
such that
It follows that
Now, we add each
F(
xi) along with its additive inverse, so that the resulting quantity is equal:
The above quantity can be written as the following sum:
Next, we employ the
mean value theorem. Stated briefly, let
F be 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
It follows that
The function
F is differentiable on the interval
[a, b]; therefore, it is also differentiable and continuous on each interval
[xi−1, xi]. According to the mean value theorem (above),
Substituting the above into (1), we get
The assumption implies
Also,
can be expressed as
of partition
.
A
converging sequence of Riemann sums. The number in the
upper left is
the total area of the blue rectangles. They converge
to the definite
integral of the function.
We are describing the area of a rectangle, with the width times the
height, and we are adding the areas together. Each rectangle, by virtue
of the
mean value theorem, describes an approximation of the curve section it is drawn over. Also
need not be the same for all values of
i, or in other words that the width of the rectangles can differ. What we have to do is approximate the curve with
n rectangles. Now, as the size of the partitions get smaller and
n increases, resulting in more partitions to cover the space, we get closer and closer to the actual area of the curve.
By taking the limit of the expression as the norm of the partitions approaches zero, we arrive at the
Riemann integral. We know that this limit exists because
f
was assumed to be integrable. That is, we take the limit as the largest
of the partitions approaches zero in size, so that all other partitions
are smaller and the number of partitions approaches infinity.
So, we take the limit on both sides of (2). This gives us
Neither
F(
b) nor
F(
a) is dependent on
, so the limit on the left side remains
F(b) − F(a).
The expression on the right side of the equation defines the integral over
f from
a to
b. Therefore, we obtain
which completes the proof.
It almost looks like the first part of the theorem follows directly from the second. That is, suppose
G is an antiderivative of
f. Then by the second theorem,
. Now, suppose
. Then
F has the same derivative as
G, and therefore
F′ = f. This argument only works, however, if we already know that
f
has an antiderivative, and the only way we know that all continuous
functions have antiderivatives is by the first part of the Fundamental
Theorem.
For example, if
f(x) = e−x2, then
f has an antiderivative, namely
and there is no simpler expression for this function. It is
therefore important not to interpret the second part of the theorem as
the definition of the integral. Indeed, there are many functions that
are integrable but lack elementary antiderivatives, and discontinuous
functions can be integrable but lack any antiderivatives at all.
Conversely, many functions that have antiderivatives are not Riemann
integrable.
Examples
As an example, suppose the following is to be calculated:
Here,
and we can use
as the antiderivative. Therefore:
Or, more generally, suppose that
is to be calculated. Here,
and
can be used as the antiderivative. Therefore:
Or, equivalently,
As a theoretical example, the theorem can be used to prove that
Since,
the result follows from,
Generalizations
We don't need to assume continuity of
f on the whole interval. Part I of the theorem then says: if
f is any
Lebesgue integrable function on
[a, b] and
x0 is a number in
[a, b] such that
f is continuous at
x0, then
is differentiable for
x = x0 with
F′(x0) = f(x0). We can relax the conditions on
f still further and suppose that it is merely locally integrable. In that case, we can conclude that the function
F is differentiable
almost everywhere and
F′(x) = f(x) almost everywhere. On the real line this statement is equivalent to
Lebesgue's differentiation theorem. These results remain true for the Henstock–Kurzweil integral, which allows a larger class of integrable functions (
Bartle 2001, Thm. 4.11).
In higher dimensions Lebesgue's differentiation theorem
generalizes the Fundamental theorem of calculus by stating that for
almost every
x, the average value of a function
f over a ball of radius
r centered at
x tends to
f(
x) as
r tends to 0.
Part II of the theorem is true for any Lebesgue integrable function
f, which has an antiderivative
F (not all integrable functions do, though). In other words, if a real function
F on
[a, b] admits a derivative
f(
x) at
every point
x of
[a, b] and if this derivative
f is Lebesgue integrable on
[a, b], then
This result may fail for continuous functions
F that admit a derivative
f(
x) at almost every point
x, as the example of the
Cantor function shows. However, if
F is
absolutely continuous, it admits a derivative
F′(
x) at almost every point
x, and moreover
F′ is integrable, with
F(b) − F(a) equal to the integral of
F′ on
[a, b]. Conversely, if
f is any integrable function, then
F as given in the first formula will be absolutely continuous with
F′ =
f a.e.
The conditions of this theorem may again be relaxed by considering the integrals involved as
Henstock–Kurzweil integrals. Specifically, if a continuous function
F(
x) admits a derivative
f(
x) at all but countably many points, then
f(
x) is Henstock–Kurzweil integrable and
F(b) − F(a) is equal to the integral of
f on
[a, b]. The difference here is that the integrability of
f does not need to be assumed. (
Bartle 2001, Thm. 4.7)
The version of
Taylor's theorem, which expresses the error term as an integral, can be seen as a generalization of the fundamental theorem.
There is a version of the theorem for
complex functions: suppose
U is an open set in
C and
f : U → C is a function that has a
holomorphic antiderivative
F on
U. Then for every curve
γ : [a, b] → U, the
curve integral can be computed as
The fundamental theorem can be generalized to curve and surface integrals in higher dimensions and on
manifolds. One such generalization offered by the
calculus of moving surfaces is the
time evolution of integrals. The most familiar extensions of the fundamental theorem of calculus in higher dimensions are the
divergence theorem and the
gradient theorem.
One of the most powerful generalizations in this direction is
Stokes' theorem (sometimes known as the fundamental theorem of multivariable calculus): Let
M be an oriented
piecewise smooth manifold of
dimension n and let
be a smooth
compactly supported (n–1)-form on
M. If ∂
M denotes the
boundary of
M given its induced
orientation, then
Here
d is the
exterior derivative, which is defined using the manifold structure only.
The theorem is often used in situations where
M is an embedded oriented submanifold of some bigger manifold (e.g.
Rk) on which the form
is defined.