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3 Integral calculus
Main article integral
Integral calculus studies methods for finding the integral of a function; which may be defined as the limit of a sum of terms, each of which corresponds to a small strip of area under the graph of a function. Considered as such, integration provides effective ways to calculate the area under a curve, and the surface area and volume of solids such as spheres and cones.
4 Foundations
The conceptual foundations of calculus include the notions of functions, limits, infinite sequences, infinite series, and continuity. Its tools include the symbol manipulation techniques associated with elementary algebra, and mathematical induction. The modern version of calculus is known as real analysis; this consists of a rigorous derivation of the results of calculus as well as generalisations such as measure theory and functional analysis.
5 Fundamental theorem of calculus
The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations. It was this realization by Newton and Leibniz that was the key to the explosion of analytic results after their work became known. This connection allows us to recover the total change in a function over some interval from its instantaneous rate of change, by integrating the latter. The fundamental theorem also provides a method to compute many definite integrals algebraically, without actually performing the limit processes, by finding antiderivatives. It also allows us to solve some differential equations, equations that relate an unknown function to its derivatives. Differential equations are ubiquitous in the sciences.
6 Applications
The development and use of calculus has had wide reaching effects on nearly all areas of modern living. It underlies nearly all of the sciences, and especially physics. Almost all modern developments such as building techniques, aviation, and nearly all other technologies make fundamental use of calculus.
Calculus has been extended to differential equations, vector calculus, calculus of variations, complex analysis, time scale calculus infinitesimal calculus, and differential topology.
7 See also
8 Further reading
- Tom M. Apostol. (1967) BooksEnthsiast.com and BooksEnthsiast.com Calculus, 2nd Ed. Wiley.
- Robert A. Adams. (1999) BooksEnthsiast.com Calculus: A complete course.
- Spivak, Michael. (Sept 1994) BooksEnthsiast.com Calculus. Publish or Perish publishing.
- Cliff Pickover. (2003) BooksEnthsiast.com Calculus and Pizza: A Math Cookbook for the Hungry Mind.
- Silvanus P. Thompson and Martin Gardner. (1998) BooksEnthsiast.com Calculus Made Easy.
- Albers, Donald J.; Richard D. Anderson and Don O. Loftsgaarden, ed. Undergraduate Programs in the Mathematics and Computer Sciences: The 1985-1986 Survey, Mathematical Association of America No. 7, 1986.
- Calculus for a New Century; A Pump, Not a Filter. Mathematical Association of America, The Association, Stony Brook, NY. 1988. ED 300 252.
- Elementary Calculus: An Approach Using Infinitesimals http://www.math.wisc.edu/~keisler/calc.html
