Calculus

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of
shape, and algebra is the study of generalizations of arithmetic operations.

Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches,
differential calculus and integral calculus. The former concerns instantaneous rates of change, and the
slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves.
These two branches are related to each other by the fundamental theorem of calculus. They make use of
the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.[1]
It is the "mathematical backbone" for dealing with problems where variables change with time or another
reference variable.[2]

Infinitesimal calculus was formulated separately in the late 17th century by Isaac Newton and Gottfried
Wilhelm Leibniz.[3][4] Later work, including codifying the idea of limits, put these developments on a
more solid conceptual footing. Today, calculus is widely used in science, engineering, biology, and even
has applications in social science and other branches of math.[5][6]

Etymology
In mathematics education, calculus is an abbreviation of both infinitesimal calculus and integral calculus,
which denotes courses of elementary mathematical analysis.

In Latin, the word calculus means “small pebble”, (the diminutive of calx, meaning "stone"), a meaning
which still persists in medicine. Because such pebbles were used for counting out distances,[7] tallying
votes, and doing abacus arithmetic, the word came to be the Latin word for calculation. In this sense, it
was used in English at least as early as 1672, several years before the publications of Leibniz and
Newton, who wrote their mathematical texts in Latin.[8]

In addition to differential calculus and integral calculus, the term is also used for naming specific
methods of computation or theories that imply some sort of computation. Examples of this usage include
propositional calculus, Ricci calculus, calculus of variations, lambda calculus, sequent calculus, and
process calculus. Furthermore, the term "calculus" has variously been applied in ethics and philosophy,
for such systems as Bentham's felicific calculus, and the ethical calculus.

History
Modern calculus was developed in 17th-century Europe by Isaac Newton and Gottfried Wilhelm Leibniz
(independently of each other, first publishing around the same time) but elements of it first appeared in
ancient Egypt and later Greece, then in China and the Middle East, and still later again in medieval
Europe and India.

Ancient precursors

Egypt
Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian Moscow
papyrus (c. 1820 BC), but the formulae are simple instructions, with no indication as to how they were
obtained.[9][10]

Greece
Laying the foundations for integral calculus and foreshadowing the
concept of the limit, ancient Greek mathematician Eudoxus of Cnidus
(c. 390–337 BC) developed the method of exhaustion to prove the
formulas for cone and pyramid volumes.

During the Hellenistic period, this method was further developed by

Archimedes (c. 287 – c. 212 BC), who combined it with a concept of the
indivisibles—a precursor to infinitesimals—allowing him to solve several
problems now treated by integral calculus. In The Method of Mechanical
Theorems he describes, for example, calculating the center of gravity of a
solid hemisphere, the center of gravity of a frustum of a circular
paraboloid, and the area of a region bounded by a parabola and one of its
secant lines.[11]

Archimedes used the

China method of exhaustion to
The method of exhaustion was later discovered independently in China by calculate the area under a
Liu Hui in the 3rd century AD to find the area of a circle.[12][13] In the 5th parabola in his work
Quadrature of the Parabola.
century AD, Zu Gengzhi, son of Zu Chongzhi, established a method[14][15]
that would later be called Cavalieri's principle to find the volume of a
sphere.

Medieval

Middle East
 In the Middle East, Hasan Ibn al-Haytham,
Latinized as Alhazen (c. 965 – c. 1040 AD)
derived a formula for the sum of fourth powers.
He used the results to carry out what would now
be called an integration of this function, where the
formulae for the sums of integral squares and
fourth powers allowed him to calculate the volume
Ibn al-Haytham, Indian mathematician and of a paraboloid.[16]
11th-century astronomer Bhāskara II
Arab
mathematician
India
and physicist Bhāskara II (c. 1114–1185) was acquainted with
some ideas of differential calculus and suggested
that the "differential coefficient" vanishes at an
extremum value of the function. [17] In his astronomical work, he gave a procedure that looked like a
precursor to infinitesimal methods. Namely, if then This can
be interpreted as the discovery that cosine is the derivative of sine. [18] In the 14th century, Indian
mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric
functions. Madhava of Sangamagrama and the Kerala School of Astronomy and Mathematics stated
components of calculus, but according to Victor J. Katz they were not able to "combine many differing
ideas under the two unifying themes of the derivative and the integral, show the connection between the
two, and turn calculus into the great problem-solving tool we have today".[16]

Modern
Johannes Kepler's work Stereometria Doliorum (1615) formed the basis of integral calculus.[19] Kepler
developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from
a focus of the ellipse.[20]

Significant work was a treatise, the origin being Kepler's methods,[20] written by Bonaventura Cavalieri,
who argued that volumes and areas should be computed as the sums of the volumes and areas of
infinitesimally thin cross-sections. The ideas were similar to Archimedes' in The Method, but this treatise
is believed to have been lost in the 13th century and was only rediscovered in the early 20th century, and
so would have been unknown to Cavalieri. Cavalieri's work was not well respected since his methods
could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.

The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite
differences developed in Europe at around the same time. Pierre de Fermat, claiming that he borrowed
from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal
error term.[21] The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter
two proving predecessors to the second fundamental theorem of calculus around 1670.[22][23]

The product rule and chain rule,[24] the notions of higher derivatives and Taylor series,[25] and of analytic
functions[26] were used by Isaac Newton in an idiosyncratic notation which he applied to solve problems
of mathematical physics. In his works, Newton rephrased his ideas to suit the mathematical idiom of the
time, replacing calculations with infinitesimals by equivalent geometrical arguments which were
considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion,
the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a
cycloid, and many other problems discussed in his Principia Mathematica (1687). In other work, he
developed series expansions for functions, including fractional and irrational powers, and it was clear that
he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time
infinitesimal methods were still considered disreputable.[27]

These ideas were arranged into a true calculus of

infinitesimals by Gottfried Wilhelm Leibniz, who
was originally accused of plagiarism by
Newton.[28] He is now regarded as an independent
inventor of and contributor to calculus. His
contribution was to provide a clear set of rules for
working with infinitesimal quantities, allowing the
computation of second and higher derivatives, and
providing the product rule and chain rule, in their
differential and integral forms. Unlike Newton,
Leibniz put painstaking effort into his choices of Gottfried Wilhelm Leibniz Isaac Newton
notation.[29] was the first to state clearly developed the use of
the rules of calculus. calculus in his laws of
Today, Leibniz and Newton are usually both given motion and universal
credit for independently inventing and developing gravitation.
calculus. Newton was the first to apply calculus to
general physics. Leibniz developed much of the
notation used in calculus today.[30]: 51–52 The basic insights that both Newton and Leibniz provided were
the laws of differentiation and integration, emphasizing that differentiation and integration are inverse
processes, second and higher derivatives, and the notion of an approximating polynomial series.

When Newton and Leibniz first published their results, there was great controversy over which
mathematician (and therefore which country) deserved credit. Newton derived his results first (later to be
published in his Method of Fluxions), but Leibniz published his "Nova Methodus pro Maximis et
Minimis" first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had
shared with a few members of the Royal Society. This controversy divided English-speaking
mathematicians from continental European mathematicians for many years, to the detriment of English
mathematics.[31] A careful examination of the papers of Leibniz and Newton shows that they arrived at
their results independently, with Leibniz starting first with integration and Newton with differentiation. It
is Leibniz, however, who gave the new discipline its name. Newton called his calculus "the science of
fluxions", a term that endured in English schools into the 19th century.[32]: 100 The first complete treatise
on calculus to be written in English and use the Leibniz notation was not published until 1815.[33]

Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing
development of calculus. One of the first and most complete works on both infinitesimal and integral
calculus was written in 1748 by Maria Gaetana Agnesi.[34][35]

Foundations
In calculus, foundations refers to the rigorous development of the subject from axioms and definitions. In
early calculus, the use of infinitesimal quantities was thought unrigorous and was fiercely criticized by
several authors, most notably Michel Rolle and Bishop Berkeley. Berkeley famously described
infinitesimals as the ghosts of departed quantities in his book The Analyst
in 1734. Working out a rigorous foundation for calculus occupied
mathematicians for much of the century following Newton and Leibniz,
and is still to some extent an active area of research today.[36]

Several mathematicians, including Maclaurin, tried to prove the soundness

of using infinitesimals, but it would not be until 150 years later when, due
to the work of Cauchy and Weierstrass, a way was finally found to avoid
mere "notions" of infinitely small quantities.[37] The foundations of
differential and integral calculus had been laid. In Cauchy's Cours
d'Analyse, we find a broad range of foundational approaches, including a Maria Gaetana Agnesi
definition of continuity in terms of infinitesimals, and a (somewhat
imprecise) prototype of an (ε, δ)-definition of limit in the definition of
differentiation.[38] In his work, Weierstrass formalized the concept of limit and eliminated infinitesimals
(although his definition can validate nilsquare infinitesimals). Following the work of Weierstrass, it
eventually became common to base calculus on limits instead of infinitesimal quantities, though the
subject is still occasionally called "infinitesimal calculus". Bernhard Riemann used these ideas to give a
precise definition of the integral.[39] It was also during this period that the ideas of calculus were
generalized to the complex plane with the development of complex analysis.[40]

In modern mathematics, the foundations of calculus are included in the field of real analysis, which
contains full definitions and proofs of the theorems of calculus. The reach of calculus has also been
greatly extended. Henri Lebesgue invented measure theory, based on earlier developments by Émile
Borel, and used it to define integrals of all but the most pathological functions.[41] Laurent Schwartz
introduced distributions, which can be used to take the derivative of any function whatsoever.[42]

Limits are not the only rigorous approach to the foundation of calculus. Another way is to use Abraham
Robinson's non-standard analysis. Robinson's approach, developed in the 1960s, uses technical
machinery from mathematical logic to augment the real number system with infinitesimal and infinite
numbers, as in the original Newton-Leibniz conception. The resulting numbers are called hyperreal
numbers, and they can be used to give a Leibniz-like development of the usual rules of calculus.[43] There
is also smooth infinitesimal analysis, which differs from non-standard analysis in that it mandates
neglecting higher-power infinitesimals during derivations.[36] Based on the ideas of F. W. Lawvere and
employing the methods of category theory, smooth infinitesimal analysis views all functions as being
continuous and incapable of being expressed in terms of discrete entities. One aspect of this formulation
is that the law of excluded middle does not hold.[36] The law of excluded middle is also rejected in
constructive mathematics, a branch of mathematics that insists that proofs of the existence of a number,
function, or other mathematical object should give a construction of the object. Reformulations of
calculus in a constructive framework are generally part of the subject of constructive analysis.[36]

Significance
While many of the ideas of calculus had been developed earlier in Greece, China, India, Iraq, Persia, and
Japan, the use of calculus began in Europe, during the 17th century, when Newton and Leibniz built on
the work of earlier mathematicians to introduce its basic principles.[13][27][44] The Hungarian polymath
John von Neumann wrote of this work,
 The calculus was the first achievement of modern mathematics and it is difficult to
overestimate its importance. I think it defines more unequivocally than anything else the
inception of modern mathematics, and the system of mathematical analysis, which is its logical
development, still constitutes the greatest technical advance in exact thinking.[45]

Applications of differential calculus include computations involving velocity and acceleration, the slope
of a curve, and optimization.[46]: 341–453 Applications of integral calculus include computations involving
area, volume, arc length, center of mass, work, and pressure.[46]: 685–700 More advanced applications
include power series and Fourier series.

Calculus is also used to gain a more precise understanding of the nature of space, time, and motion. For
centuries, mathematicians and philosophers wrestled with paradoxes involving division by zero or sums
of infinitely many numbers. These questions arise in the study of motion and area. The ancient Greek
philosopher Zeno of Elea gave several famous examples of such paradoxes. Calculus provides tools,
especially the limit and the infinite series, that resolve the paradoxes.[47]

Principles

Limits and infinitesimals

Calculus is usually developed by working with very small quantities. Historically, the first method of
doing so was by infinitesimals. These are objects which can be treated like real numbers but which are, in
some sense, "infinitely small". For example, an infinitesimal number could be greater than 0, but less than
any number in the sequence 1, 1/2, 1/3, ... and thus less than any positive real number. From this point of
view, calculus is a collection of techniques for manipulating infinitesimals. The symbols and were
taken to be infinitesimal, and the derivative was their ratio. [36]

The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion
of an infinitesimal precise. In the late 19th century, infinitesimals were replaced within academia by the
epsilon, delta approach to limits. Limits describe the behavior of a function at a certain input in terms of
its values at nearby inputs. They capture small-scale behavior using the intrinsic structure of the real
number system (as a metric space with the least-upper-bound property). In this treatment, calculus is a
collection of techniques for manipulating certain limits. Infinitesimals get replaced by sequences of
smaller and smaller numbers, and the infinitely small behavior of a function is found by taking the
limiting behavior for these sequences. Limits were thought to provide a more rigorous foundation for
calculus, and for this reason, they became the standard approach during the 20th century. However, the
infinitesimal concept was revived in the 20th century with the introduction of non-standard analysis and
smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals.[36]

Differential calculus
Differential calculus is the study of the definition, properties, and applications of the derivative of a
function. The process of finding the derivative is called differentiation. Given a function and a point in
the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near
that point. By finding the derivative of a function at every point in its domain, it is possible to produce a
new function, called the derivative function or just the
derivative of the original function. In formal terms,
the derivative is a linear operator which takes a
function as its input and produces a second function
as its output. This is more abstract than many of the
processes studied in elementary algebra, where
functions usually input a number and output another
number. For example, if the doubling function is
given the input three, then it outputs six, and if the
squaring function is given the input three, then it
outputs nine. The derivative, however, can take the
Tangent line at (x0, f(x0)). The derivative f′(x) of a
squaring function as an input. This means that the curve at a point is the slope (rise over run) of the
derivative takes all the information of the squaring line tangent to that curve at that point.
function—such as that two is sent to four, three is sent
to nine, four is sent to sixteen, and so on—and uses
this information to produce another function. The function produced by differentiating the squaring
function turns out to be the doubling function.[30]: 32

In more explicit terms the "doubling function" may be denoted by g(x) = 2x and the "squaring function"
by f(x) = x2. The "derivative" now takes the function f(x), defined by the expression "x2", as an input,
that is all the information—such as that two is sent to four, three is sent to nine, four is sent to sixteen,
and so on—and uses this information to output another function, the function g(x) = 2x, as will turn out.

In Lagrange's notation, the symbol for a derivative is an apostrophe-like mark called a prime. Thus, the
derivative of a function called f is denoted by f′, pronounced "f prime" or "f dash". For instance, if
f(x) = x2 is the squaring function, then f′(x) = 2x is its derivative (the doubling function g from above).

If the input of the function represents time, then the derivative represents change concerning time. For
example, if f is a function that takes time as input and gives the position of a ball at that time as output,
then the derivative of f is how the position is changing in time, that is, it is the velocity of the
ball.[30]: 18–20

If a function is linear (that is if the graph of the function is a straight line), then the function can be
written as y = mx + b, where x is the independent variable, y is the dependent variable, b is the y-
intercept, and:

This gives an exact value for the slope of a straight line.[48]: 6 If the graph of the function is not a straight
line, however, then the change in y divided by the change in x varies. Derivatives give an exact meaning
to the notion of change in output concerning change in input. To be concrete, let f be a function, and fix a
point a in the domain of f. (a, f(a)) is a point on the graph of the function. If h is a number close to zero,
then a + h is a number close to a. Therefore, (a + h, f(a + h)) is close to (a, f(a)). The slope between
these two points is
This expression is called a difference quotient. A line through two points on a curve is called a secant line,
so m is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). The second line is only an
approximation to the behavior of the function at the point a because it does not account for what happens
between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this
would require dividing by zero, which is undefined. The derivative is defined by taking the limit as h
tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent
value for the case when h equals zero:

Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a
limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative
is sometimes called the slope of the function f.[48]: 61–63

Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x2 be the
squaring function.

The slope of the tangent line to the squaring function

at the point (3, 9) is 6, that is to say, it is going up six
times as fast as it is going to the right. The limit
process just described can be performed for any point
in the domain of the squaring function. This defines
The derivative f′(x) of a curve at a point is the
the derivative function of the squaring function or just
slope of the line tangent to that curve at that point.
the derivative of the squaring function for short. A This slope is determined by considering the
computation similar to the one above shows that the limiting value of the slopes of the second lines.
derivative of the squaring function is the doubling Here the function involved (drawn in red) is
function.[48]: 63 f(x) = x3 − x. The tangent line (in green) which
passes through the point (−3/2, −15/8) has a
slope of 23/4. The vertical and horizontal scales in
Leibniz notation this image are different.
A common notation, introduced by Leibniz, for the
derivative in the example above is
 dy
In an approach based on limits, the symbol ⁠ ⁠is to be interpreted not as the quotient of two numbers but
dx
as a shorthand for the limit computed above.[48]: 74 Leibniz, however, did intend it to represent the
quotient of two infinitesimally small numbers, dy being the infinitesimally small change in y caused by
d
an infinitesimally small change dx applied to x. We can also think of ⁠ ⁠ as a differentiation operator,
dx
which takes a function as an input and gives another function, the derivative, as the output. For example:

In this usage, the dx in the denominator is read as "with respect to x".[48]: 79 Another example of correct
notation could be:

Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate
symbols like dx and dy as if they were real numbers; although it is possible to avoid such manipulations,
they are sometimes notationally convenient in expressing operations such as the total derivative.

Integral calculus
Integral calculus is the study of the definitions, properties, and applications of two related concepts, the
indefinite integral and the definite integral. The process of finding the value of an integral is called
integration.[46]: 508 The indefinite integral, also known as the antiderivative, is the inverse operation to
the derivative.[48]: 163–165 F is an indefinite integral of f when f is a derivative of F. (This use of lower-
and upper-case letters for a function and its indefinite integral is common in calculus.) The definite
integral inputs a function and outputs a number, which gives the algebraic sum of areas between the
graph of the input and the x-axis. The technical definition of the definite integral involves the limit of a
sum of areas of rectangles, called a Riemann sum.[49]: 282

A motivating example is the distance traveled in a given time.[48]: 153 If the speed is constant, only
multiplication is needed:

But if the speed changes, a more powerful method of finding the distance is necessary. One such method
is to approximate the distance traveled by breaking up the time into many short intervals of time, then
multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum
(a Riemann sum) of the approximate distance traveled in each interval. The basic idea is that if only a
short time elapses, then the speed will stay more or less the same. However, a Riemann sum only gives an
approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exact
distance traveled.

When velocity is constant, the total distance traveled over the given time interval can be computed by
multiplying velocity and time. For example, traveling a steady 50 mph for 3 hours results in a total
distance of 150 miles. Plotting the velocity as a function of time yields a rectangle with a height equal to
the velocity and a width equal to the time elapsed.
Therefore, the product of velocity and time also
calculates the rectangular area under the (constant)
velocity curve.[46]: 535 This connection between the
area under a curve and the distance traveled can be
extended to any irregularly shaped region exhibiting a
fluctuating velocity over a given period. If f(x)
represents speed as it varies over time, the distance
traveled between the times represented by a and b is
the area of the region between f(x) and the x-axis,
between x = a and x = b.

To approximate that area, an intuitive method would be

to divide up the distance between a and b into several
Integration can be thought of as measuring the
equal segments, the length of each segment represented area under a curve, defined by f(x), between
by the symbol Δx. For each small segment, we can two points (here a and b).
choose one value of the function f(x). Call that value
h. Then the area of the rectangle with base Δx and
height h gives the distance (time Δx multiplied by
speed h) traveled in that segment. Associated with each
segment is the average value of the function above it,
f(x) = h. The sum of all such rectangles gives an
approximation of the area between the axis and the
curve, which is an approximation of the total distance
A sequence of midpoint Riemann sums over a
traveled. A smaller value for Δx will give more
regular partition of an interval: the total area of
rectangles and in most cases a better approximation,
the rectangles converges to the integral of the
but for an exact answer, we need to take a limit as Δx function.
approaches zero.[46]: 512–522

The symbol of integration is , an elongated S chosen to suggest summation.[46]: 529 The definite integral

is written as:

and is read "the integral from a to b of f-of-x with respect to x." The Leibniz notation dx is intended to
suggest dividing the area under the curve into an infinite number of rectangles so that their width Δx
becomes the infinitesimally small dx.[30]: 44

The indefinite integral, or antiderivative, is written:

Functions differing by only a constant have the same derivative, and it can be shown that the
antiderivative of a given function is a family of functions differing only by a constant.[49]: 326 Since the
derivative of the function y = x2 + C, where C is any constant, is y′ = 2x, the antiderivative of the latter
is given by:

The unspecified constant C present in the indefinite integral or antiderivative is known as the constant of
integration.[50]: 135

Fundamental theorem
The fundamental theorem of calculus states that differentiation and integration are inverse
operations.[49]: 290 More precisely, it relates the values of antiderivatives to definite integrals. Because it
is usually easier to compute an antiderivative than to apply the definition of a definite integral, the
fundamental theorem of calculus provides a practical way of computing definite integrals. It can also be
interpreted as a precise statement of the fact that differentiation is the inverse of integration.

The fundamental theorem of calculus states: If a function f is continuous on the interval [a, b] and if F is
a function whose derivative is f on the interval (a, b), then

Furthermore, for every x in the interval (a, b),

This realization, made by both Newton and Leibniz, was key to the proliferation of analytic results after
their work became known. (The extent to which Newton and Leibniz were influenced by immediate
predecessors, and particularly what Leibniz may have learned from the work of Isaac Barrow, is difficult
to determine because of the priority dispute between them.[51]) The fundamental theorem provides an
algebraic method of computing many definite integrals—without performing limit processes—by finding
formulae for antiderivatives. It is also a prototype solution of a differential equation. Differential
equations relate an unknown function to its derivatives and are ubiquitous in the sciences.[52]: 351–352

Applications
Calculus is used in every branch of the physical sciences,[53]: 1 actuarial science, computer science,
statistics, engineering, economics, business, medicine, demography, and in other fields wherever a
problem can be mathematically modeled and an optimal solution is desired.[54] It allows one to go from
(non-constant) rates of change to the total change or vice versa, and many times in studying a problem we
know one and are trying to find the other.[55] Calculus can be used in conjunction with other
mathematical disciplines. For example, it can be used with linear algebra to find the "best fit" linear
approximation for a set of points in a domain. Or, it can be used in probability theory to determine the
expectation value of a continuous random variable given a probability density function.[56]: 37 In analytic
geometry, the study of graphs of functions, calculus is used to find high points and low points (maxima
and minima), slope, concavity and inflection points. Calculus is also used to find approximate solutions
to equations; in practice, it is the standard way to solve differential
equations and do root finding in most applications. Examples are
methods such as Newton's method, fixed point iteration, and linear
approximation. For instance, spacecraft use a variation of the
Euler method to approximate curved courses within zero-gravity
environments.

Physics makes particular use of calculus; all concepts in classical

mechanics and electromagnetism are related through calculus. The
The logarithmic spiral of the nautilus
mass of an object of known density, the moment of inertia of
shell is a classical image used to
objects, and the potential energies due to gravitational and depict the growth and change
electromagnetic forces can all be found by the use of calculus. An related to calculus.
example of the use of calculus in mechanics is Newton's second
law of motion, which states that the derivative of an object's
momentum concerning time equals the net force upon it. Alternatively, Newton's second law can be
expressed by saying that the net force equals the object's mass times its acceleration, which is the time
derivative of velocity and thus the second time derivative of spatial position. Starting from knowing how
an object is accelerating, we use calculus to derive its path.[57]

Maxwell's theory of electromagnetism and Einstein's theory of general relativity are also expressed in the
language of differential calculus.[58][59]: 52–55 Chemistry also uses calculus in determining reaction
rates[60]: 599 and in studying radioactive decay.[60]: 814 In biology, population dynamics starts with
reproduction and death rates to model population changes.[61][62]: 631

Green's theorem, which gives the relationship between a line integral around a simple closed curve C and
a double integral over the plane region D bounded by C, is applied in an instrument known as a
planimeter, which is used to calculate the area of a flat surface on a drawing.[63] For example, it can be
used to calculate the amount of area taken up by an irregularly shaped flower bed or swimming pool
when designing the layout of a piece of property.

In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel to
maximize flow.[64] Calculus can be applied to understand how quickly a drug is eliminated from a body
or how quickly a cancerous tumor grows.[65]

In economics, calculus allows for the determination of maximal profit by providing a way to easily
calculate both marginal cost and marginal revenue.[66]: 387

See also
Glossary of calculus
List of calculus topics
List of derivatives and integrals in alternative calculi
List of differentiation identities
Publications in calculus
 Table of integrals

References
1. DeBaggis, Henry F.; Miller, Kenneth S. (1966). Foundations of the Calculus. Philadelphia:
Saunders. OCLC 527896 (https://search.worldcat.org/oclc/527896).
2. Fox, Huw; Bolton, Bill (2002), "Calculus" (https://linkinghub.elsevier.com/retrieve/pii/B97807
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global ed.). Harlow: Pearson. ISBN 978-1-292-15446-6. OCLC 1064041906 (https://search.
worldcat.org/oclc/1064041906).

Further reading
Adams, Robert A. (1999). Calculus: A complete course. Addison-Wesley. ISBN 978-0-201-
39607-2.
Albers, Donald J.; Anderson, Richard D.; Loftsgaarden, Don O., eds. (1986). Undergraduate
Programs in the Mathematics and Computer Sciences: The 1985–1986 Survey.
Mathematical Association of America.
Anton, Howard; Bivens, Irl; Davis, Stephen (2002). Calculus. John Wiley and Sons Pte. Ltd.
ISBN 978-81-265-1259-1.
Apostol, Tom M. (1967). Calculus, Volume 1, One-Variable Calculus with an Introduction to
Linear Algebra. Wiley. ISBN 978-0-471-00005-1.
Apostol, Tom M. (1969). Calculus, Volume 2, Multi-Variable Calculus and Linear Algebra
with Applications. Wiley. ISBN 978-0-471-00007-5.
Bell, John Lane (1998). A Primer of Infinitesimal Analysis. Cambridge University Press.
ISBN 978-0-521-62401-5. Uses synthetic differential geometry and nilpotent infinitesimals.
Boelkins, M. (2012). Active Calculus: a free, open text (https://web.archive.org/web/2013053
0024317/http://faculty.gvsu.edu/boelkinm/Home/Download_files/Active%20Calculus%20ch1
-8%20%28v.1.1%20W13%29.pdf) (PDF). Archived from the original (http://gvsu.edu/s/km)
on 30 May 2013. Retrieved 1 February 2013.
Boyer, Carl Benjamin (1959) [1949]. The History of the Calculus and its Conceptual
Development (https://books.google.com/books?id=KLQSHUW8FnUC) (Dover ed.). Hafner.
ISBN 0-486-60509-4.
Cajori, Florian (September 1923). "The History of Notations of the Calculus". Annals of
Mathematics. 2nd Series. 25 (1): 1–46. doi:10.2307/1967725 (https://doi.org/10.2307%2F19
67725). hdl:2027/mdp.39015017345896 (https://hdl.handle.net/2027%2Fmdp.39015017345
896). JSTOR 1967725 (https://www.jstor.org/stable/1967725).
Courant, Richard (3 December 1998). Introduction to calculus and analysis 1. Springer.
ISBN 978-3-540-65058-4.
Gonick, Larry (2012). The Cartoon Guide to Calculus. William Morrow. ISBN 978-0-061-
68909-3. OCLC 932781617 (https://search.worldcat.org/oclc/932781617).
 Keisler, H.J. (2000). Elementary Calculus: An Approach Using Infinitesimals. Retrieved 29
August 2010 from http://www.math.wisc.edu/~keisler/calc.html (http://www.math.wisc.edu/~k
eisler/calc.html) Archived (https://web.archive.org/web/20110501113944/http://www.math.wi
sc.edu/~keisler/calc.html) 1 May 2011 at the Wayback Machine
Landau, Edmund (2001). Differential and Integral Calculus. American Mathematical Society.
ISBN 0-8218-2830-4.
Lebedev, Leonid P.; Cloud, Michael J. (2004). "The Tools of Calculus". Approximating
Perfection: a Mathematician's Journey into the World of Mechanics. Princeton University
Press. Bibcode:2004apmj.book.....L (https://ui.adsabs.harvard.edu/abs/2004apmj.book.....L).
Larson, Ron; Edwards, Bruce H. (2010). Calculus (9th ed.). Brooks Cole Cengage Learning.
ISBN 978-0-547-16702-2.
McQuarrie, Donald A. (2003). Mathematical Methods for Scientists and Engineers.
University Science Books. ISBN 978-1-891389-24-5.
Pickover, Cliff (2003). Calculus and Pizza: A Math Cookbook for the Hungry Mind. John
Wiley & Sons. ISBN 978-0-471-26987-8.
Salas, Saturnino L.; Hille, Einar; Etgen, Garret J. (2007). Calculus: One and Several
Variables (10th ed.). Wiley. ISBN 978-0-471-69804-3.
Spivak, Michael (September 1994). Calculus. Publish or Perish publishing. ISBN 978-0-
914098-89-8.
Steen, Lynn Arthur, ed. (1988). Calculus for a New Century; A Pump, Not a Filter.
Mathematical Association of America. ISBN 0-88385-058-3.
Stewart, James (2012). Calculus: Early Transcendentals (7th ed.). Brooks Cole Cengage
Learning. ISBN 978-0-538-49790-9.
Thomas, George Brinton; Finney, Ross L.; Weir, Maurice D. (1996). Calculus and Analytic
Geometry, Part 1. Addison Wesley. ISBN 978-0-201-53174-9.
Thomas, George B.; Weir, Maurice D.; Hass, Joel; Giordano, Frank R. (2008). Calculus
(11th ed.). Addison-Wesley. ISBN 978-0-321-48987-6.
Thompson, Silvanus P.; Gardner, Martin (1998). Calculus Made Easy. Macmillan. ISBN 978-
0-312-18548-0.

External links
"Calculus" (https://www.encyclopediaofmath.org/index.php?title=Calculus), Encyclopedia of
Mathematics, EMS Press, 2001 [1994]
Weisstein, Eric W. "Calculus" (https://mathworld.wolfram.com/Calculus.html). MathWorld.
Topics on Calculus (https://planetmath.org/TopicsOnCalculus) at PlanetMath.
Calculus Made Easy (1914) by Silvanus P. Thompson (http://djm.cc/library/Calculus_Made_
Easy_Thompson.pdf) Full text in PDF
Calculus (https://www.bbc.co.uk/programmes/b00mrfwq) on In Our Time at the BBC
Calculus.org: The Calculus page (http://www.calculus.org) at University of California,
Davis – contains resources and links to other sites
Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis (http://ww
w.economics.soton.ac.uk/staff/aldrich/Calculus%20and%20Analysis%20Earliest%20Uses.ht
m)
The Role of Calculus in College Mathematics (http://www.ericdigests.org/pre-9217/calculus.
htm) Archived (https://web.archive.org/web/20210726234750/http://www.ericdigests.org/pre-
9217/calculus.htm) 26 July 2021 at the Wayback Machine from ERICDigests.org
OpenCourseWare Calculus (https://ocw.mit.edu/courses/mathematics/18-01sc-single-variabl
e-calculus-fall-2010/) from the Massachusetts Institute of Technology
 Infinitesimal Calculus (http://www.encyclopediaofmath.org/index.php?title=Infinitesimal_calc
ulus&oldid=18648) – an article on its historical development, in Encyclopedia of
Mathematics, ed. Michiel Hazewinkel.
Daniel Kleitman, MIT. "Calculus for Beginners and Artists" (http://math.mit.edu/~djk/calculus
_beginners/).
Calculus training materials at imomath.com (http://www.imomath.com/index.php?options=27
7)
(in English and Arabic) The Excursion of Calculus (http://www.wdl.org/en/item/4327/), 1772

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