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Topic: Integral calculus


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  Calculus - Wikipedia, the free encyclopedia
Though the origins of integral calculus are generally regarded as going back no farther than to the time of the ancient Greeks, circa 200 B.C., there is some evidence that the ancient Egyptians may have had some hint of the idea at a much earlier date.
The rigorous foundation of calculus is based on the notions of a function and of a limit; the latter has a theory ultimately depending on that of the real numbers as a continuum.
A Brief Introduction to Infinitesimal Calculus by Keith Duncan Stroyan of the University of Iowa.
en.wikipedia.org /wiki/Calculus   (2183 words)

  
 Integral - Wikipedia, the free encyclopedia
In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total.
Improper integrals usually turn up when the range of the function to be integrated is infinite or, in the case of the Riemann integral, when the domain of the function is infinite.
The Riemann-Stieltjes integral, an extension of the Riemann integral.
en.wikipedia.org /wiki/Integral   (1468 words)

  
 Calculus - Encyclopedia.WorldSearch   (Site not responding. Last check: 2007-10-21)
Though the origins of integral calculus are generally regarded as going no farther back than to the ancient Greeks, there is evidence that the ancient Egyptians may have harbored such knowledge amongst themselves as well.
Integral calculus determines the exact distance traveled during an interval of time by creating a series of better and better approximations, called Riemann sums, that approach the exact speed.
Calculus has been extended to differential equations, vector calculus, calculus of variations, complex analysis, time scale calculus, infinitesimal calculus, and differential topology.
encyclopedia.worldsearch.com /calculus.htm   (1750 words)

  
 calculus -> The Integral Calculus on Encyclopedia.com 2002   (Site not responding. Last check: 2007-10-21)
The second important kind of limit encountered in the calculus is the limit of a sum of elements when the number of such elements increases without bound while the size of the elements diminishes.
This connection between the integral and the derivative is known as the Fundamental Theorem of the Calculus.
The branch of calculus concerned with both the integral as the limit of a sum and the integral as the antiderivative of a function is known as the integral calculus.
www.encyclopedia.com /html/section/calcul_theintegralcalculus.asp   (966 words)

  
 [No title]   (Site not responding. Last check: 2007-10-21)
The integral value, of a real number x, is defined as the largest integer which is less than, or equal to, x; this is often denoted by ; known as the "floor function".
In calculus, the integral, of a function, is the size of the area bounded by the x-axis and the graph of a function, f(x); negative areas are possible.
This approach is motivated by calculus, and is the main method used for calculating the area under the curve as described in the preceding paragraph, for functions given by formulae.
www.informationgenius.com /encyclopedia/i/in/integral.html   (748 words)

  
 Integral   (Site not responding. Last check: 2007-10-21)
The integral value of a real number x is defined as the largest integer which is less than, or equal to, x.
An integral which can only be evaluated by considering it as the limit of integrals on successively larger and larger integrals is called an improper integral.
Improper integrals usually turn up when the range of the function is infinite or, in the case of the Riemann integral, when the domain is infinite.
www.sciencedaily.com /encyclopedia/integral   (1529 words)

  
 Math.com Online Solvers Calculus   (Site not responding. Last check: 2007-10-21)
Calculus is a vast topic, and it forms the basis for much of modern mathematics.
At school, you are introduced to differential calculus by learning how to find the derivative of a function in order to determine the slope of the graph of that function at any point.
Integral calculus is often introduced in school in terms of finding primitive functions (indefinite integrals) and finding the area under a curve (definite integrals).
www.math.com /students/solvers/calculus/calculus.htm   (230 words)

  
 Non-Newtonian Calculus
Each non-Newtonian calculus, as well as the classical calculus, can be ‘weighted’ in a manner explained in the book "The First Systems of Weighted Differential and Integral Calculus” (QA303.G876) by Jane Grossman, Michael Grossman, and Robert Katz.
In his article "Non-Newtonian Calculus Applied to Probability, Utility, and Bayesian Analysis" (Proceedings of the American Statistical Association, 1980), he used non-Newtonian calculus to create a theory of probability that is adapted to human behavior and decision making.
That a whole family of differential and integral calculi, parallel to but nonlinear with respect to ordinary Newtonian (or Leibnizian) calculus, should have remained undiscovered (or uninvented) for so long is astonishing -- but true.
www.geocities.com /nonnewtoniancalculus   (682 words)

  
 BBC Education - AS Guru - Maths - Methods - Integration
The beginnings of integral calculus were laid in a treatise of Archimedes (225 BCE).
It was on October 29th 1675 that he first used the modern integral symbol as a long "s" derived from the initial of the Latin summa (sum).
Much of the use of the calculus in the Eighteenth Century was applied in the areas of Mechanics and Astronomy.
www.bbc.co.uk /education/asguru/maths/12methods/04integration/18integration/index.shtml   (302 words)

  
 Gottfried Wilhelm Leibnitz (1646 - 1716)
The development of that calculus was the main work of the mathematicians of the first half of the eighteenth century.
In both of these papers the principle of continuity is explicitly assumed, while his treatment of the subject is based on the use of infinitesimals and not on that of the limiting value of ratios.
But in spite of this, his title to fame rests on a sure basis, for by his advocacy of the differential calculus his name is inseparably connected with one of the chief instruments of analysis, as that of Descartes - another philosopher - is similarly connected with analytical geometry.
www.maths.tcd.ie /pub/HistMath/People/Leibniz/RouseBall/RB_Leibnitz.html   (2618 words)

  
 World Web Math: Calculus Summary   (Site not responding. Last check: 2007-10-21)
Differential calculus studies the derivative and integral calculus studies (surprise!) the integral.
The derivative and integral are linked in that they are both defined via the concept of the limit: they are inverse operations of each other (a fact sometimes known as the fundamental theorem of calculus): and they are both fundamental to much of modern science as we know it.
These are improper integrals, and can be found by taking the limit of an integral over an interval that either grows towards infinity or towards the points where the function is undefined.
web.mit.edu /wwmath/calculus/summary.html   (911 words)

  
 Distance Calculus at Suffolk University
The Distance Calculus program at Suffolk University was founded in 1997 with one of the instructors from The Ohio State University program.
Distance Calculus at Suffolk University is led by Dr. Robert Curtis, a member of the faculty of the Mathematics and Computer Science Department at Suffolk University since 1996.
Distance Calculus is operated by a team of educators through MathMonkeys, LLC, publishers of LiveMath and MathEQ software, based in Harvard Square, Cambridge, Massachusetts, USA.
www.distancecalculus.com   (682 words)

  
 calculus.org - THE CALCULUS PAGE .
Also available are scanned solutions to problems in differential, integral and multi-variable calculus and series.
Calculus Lecture Notes from the University of Toronto at Scarsborough.
Single Variable Calculus Mika Seppälä of Florida State University and the University of Helsinki presents classroom type notes on calculus, in pdf and powerpoint format.
www.calculus.org   (1134 words)

  
 Calculus graphics -- Douglas N. Arnold
The diagram illustrates the local accuracy of the tangent line approximation to a smooth curve, or--otherwise stated--the closeness of the differential of a function to the difference of function values due to a small increment of the independent variable.
The proof is based on a diagram depicting a circular sector in the unit circle together with an inscribed and a circumscribed triangle.
A brief graphical exploration of a continuous, nowhere differentiable function fits very well in the first semester of calculus, for example, to provide a strong counterexample to the converse of the theorem that differentiability implies continuity; or to show that it is only differentiable functions which look like straight lines under the microscope.
www.math.psu.edu /dna/graphics.html   (1432 words)

  
 Visual Calculus - Fundamental Theorem of Calculus   (Site not responding. Last check: 2007-10-21)
is an indefinite integral or antiderivative of f.
Illustration of the Fundamental Theorem of Calculus using Maple and a LiveMath Notebook.
LiveMath Notebook which evaluates the derivative of a function which is an integral with variable limits.
archives.math.utk.edu /visual.calculus/4/ftc.9   (232 words)

  
 Math Forum - Ask Dr. Math
INTEGRAL sin^n(x)*dx = INTEGRAL u*dv, = u*v - INTEGRAL v*du.
Now bring the least term on the right over to the left and divide by n: INTEGRAL sin^n(x)*dx = -(1/n)*sin^(n-1)(x)*cos(x) + ((n-1)/n)*INTEGRAL sin^(n-2)(x)*dx This formula lets you reduce the exponent from 10 to 8, then to 6, then to 4, then to 2, and finally to INTEGRAL dx, which is easy.
If instead you meant that the expression to be integrated to be (1/x)*(1+x^4)^(3/4), then this is one of those functions that cannot be integrated in closed form in terms of the familiar functions of calculus.
mathforum.org /library/drmath/view/52094.html   (354 words)

  
 Calculus history
In fact, although Barrow never explicitly stated the fundamental theorem of the calculus, he was working towards the result and Newton was to continue with this direction and state the Fundamental Theorem of the Calculus explicitly.
His results on the integral calculus were published in 1684 and 1686 under the name 'calculus summatorius', the name integral calculus was suggested by Jacob Bernoulli in 1690.
After Newton and Leibniz the development of the calculus was continued by Jacob Bernoulli and Johann Bernoulli.
www-gap.dcs.st-and.ac.uk /~history/HistTopics/The_rise_of_calculus.html   (1695 words)

  
 Integral Calculus Made Easy
Tim Mooney ===================================================== Integral calculus is essentially looking for the area made by a curve in a certain coordinate system.
Let us start with an easier question, "What is differential calculus?" Differential calculus is a set of mathematical theorems that allows one to find the slope of a curve at any point on the curve, provided certain conditions are met.
Without going into the "plumbing" you can say that finding the areas, and volumes of various shapes is done using integral calculus, as well as the length of curves (pieces of string that are curved), the surface areas of various shapes.
www.newton.dep.anl.gov /askasci/math99/math99173.htm   (624 words)

  
 Northeastern University, Department of Mathematics
The integral calculus is applied to accumulation functions and future value.
Serves as both the first half of a two semester calculus sequence and as a self-contained one semester coure in differential and integral calculus.
Starting with the algebra and geometry of complex numbers, basic derivative and contour integral properties are developed for elementary algebraic and transcendental functions as well as for other analytic functions and functions with isolated singularities.
www.math.neu.edu /undergrad/ugcatalog.html   (5966 words)

  
 The Fundamental Theorem of Integral Calculus   (Site not responding. Last check: 2007-10-21)
Theorem 4914 (The Fundamental Theorem of Integral Calculus)
The Preliminary Fundamental Theorem of Calculus shows that F(b) - F(a) always lies between the upper and lower Riemann sums, so F(b)-F(a) must be the integral of f on [a,b].
The three Theorems: The Fundamental Theorem of Integral Calculus, Limits of Riemann Sums, and Integrable Functions, make up the heart most applications of integration.
www.uwm.edu /~ericskey/226S99/CLN/node29.html   (212 words)

  
 Alex Suciu: Integral Calculus
Also introduced are the calculus of trigonometric and exponential functions, and methods of solving simple differential equations.
Integrals as areas and sums, the Fundamental Theorem of Calculus, properties of the integral, applications (average value, volume).
What they are, calculus of trig functions, inverse trig functions.
www.math.neu.edu /~suciu/mth1108/intcalc.sp99.html   (174 words)

  
 Mathematics Archives Calculus Resources On-Line
In addition to the "calculus" directories in the Mac and Windows/MS-DOS areas, interesting programs may be found elsewhere (e.g., in the directories "Advanced Calculus", "Graphing Programs", etc.).
1989 and 1993) is developing and disseminating an innovative core calculus curriculum intended to be practical and attractive to a multitude of institutions.
SimCalc: Simulations for Calculus Learning is a project to build and test a series of software simulations and curriculum materials designed to support learning of the underlying ideas of calculus by mainstream students in grades 3-12.
archives.math.utk.edu /calculus/crol.html   (1852 words)

  
 integral calculus: essaysdoctor.com- the essays, term papers, research papers doctor
This dynamic makes sense ethically, since businesses are an integral part of the communities in which they live, and should therefore be accountable to the people that they serve and their respective interests.
essaysdoctor.com is a website that has a wealth of free essay abstracts on integral calculus.
If you feel that the abstract matches what you're looking for, you can download the integral calculus abstract directly from essaysdoctor.com.
www.essaysdoctor.com /term-papers/401482/integral-calculus.html   (326 words)

  
 Definite Integrals - 1   (Site not responding. Last check: 2007-10-21)
How to use Maple to evaluate a definite integral using the definition.
How to use Derive to evaluate a definite integral using the definition.
A complicated example of evaluating definite integrals using the definition.
archives.math.utk.edu /visual.calculus/4/definite.1/index.html   (65 words)

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