| Topics in calculus | | Fundamental theorem | Function | Limits of functions | Continuity | Calculus with polynomials | Mean value theorem | Vector calculus | Tensor calculus For other uses of the term calculus see calculus (disambiguation) Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ...
Part I It is given that Let there be two numbers x1 and x1 + Δx in [a, b]. So we have and . Subtracting the two equations gives . It can be shown that . (The sum of the areas of two adjacent regions is equal to the area of both regions combined. ...
In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...
In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ...
In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...
In mathematics, polynomials are perhaps the simplest functions with which to do calculus. ...
For any function that is continuous on [a, b] and differentiable on (a, b) there exists some c in the interval (a, b) such that the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c. ...
Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. ...
In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalar, vector and linear operator. ...
| | Differentiation | | Product rule | Quotient rule | Chain rule | Implicit differentiation | Taylor's theorem | Related rates The derivative in mathematics (specifically, differential calculus) is a quantity that measures, on continuous functions, the limit of a rate of change, , as approaches 0. ...
In mathematics, the product rule of calculus, which is also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...
In calculus, the quotient rule is a method of finding the derivative of a function which is the quotient of two other functions for which derivatives exist. ...
In calculus, the chain rule is a formula for the derivative of the composition of two functions. ...
In mathematics, to give an implicit function f is to give the graph of a function, as a relation. ...
In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ...
In differential calculus, related rates problems involve ratios of derivatives of two or more related variables that are changing with respect to time. ...
| | Integration | | Integration by substitution | Integration by parts | Integration by trigonometric substitution | Solids of revolution | Integration by disks | Integration by cylindrical shells | Improper integrals | Lists of integrals In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ...
In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ...
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...
In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. ...
In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis) that lies on the same plane. ...
In mathematics, in particular integral calculus, disk integration (the disk method) is a means of calculating the volume of a solid of revolution. ...
It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ...
See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of hyperbolic functions List of integrals of exponential functions List of integrals of logarithmic functions List of integrals of arc functions...
| Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution. This article deals with the concept of an integral in calculus. ...
A calculation is a deliberate process for transforming one or more inputs into one or more results. ...
Volume (also called capacity) is a quantification of how much space an object occupies. ...
In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis) that lies on the same plane. ...
It makes use of the so-called "representative cylinder". Intuitively speaking, part of the graph of a function is rotated around an axis, and is modelled by an infinite number of hollow pipes, all infinitely thin. The word cylinder has several meanings. ...
In mathematics, the graph of a function f(x1, x2, ..., xn) is the collection of all tuples (x1, x2, ..., xn, f(x1, ..., xn)). In particular, graph means the graphical representation of this collection, in the form of a curve or surface, together with axes, etc. ...
The axis of rotation of a rotating body is a line such that the distance between any point on the line and any point of the body remains constant under the rotation. ...
The idea is that a "representative rectangle" (used in the most basic forms of integration -- such as ∫ x dx) can be rotated about the axis of revolution; thus generating a hollow cylinder. Integration, as an accumulative process, can then calculate the integrated volume of a "family" of shells (a shell being the outer edge of a hollow cylinder) -- as volume is the antiderivative of area, if one can calculate the lateral surface area of a shell...one can then calculate its volume. In geometry, a rectangle is a defined as a quadrilateral polygon in which all four angles are right angles. ...
In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ...
In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis) that lies on the same plane. ...
A family of Ouagadougou, Burkina Faso in 1997 A family is a domestic group of people, or a number of domestic groups affiliated by blood or by a variety of legal ties such as marriage, domestic partnership, adoption, surname and in some cases slavery as was the case in the...
In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ...
The necessary equation, for calculating such a volume, V, is slightly different depending on which axis is serving as the axis of revolution. These equations note that the lateral surface area of a shell equals: 2 pi (π) multiplied by the cylinder's average radius, p(y), multiplied by the length of the cylinder, h(y). One can calculate the volume of a representative shell by: 2π * p(y) * h(y) * dy, where dy is the thickness of the shell -- that being some number approaching zero. In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ...
See also the disambiguation page title equality. ...
The minuscule, or lower-case, pi The mathematical constant π represents the ratio of a circles circumference to its diameter and is commonly used in mathematics, physics, and engineering. ...
In its simplest form, multiplication is a quick way of adding identical numbers. ...
RADIUS (Remote Authentication Dial In User Service) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ...
In general English usage, length (symbols: l, L) is but one particular instance of distance – an objects length is how long the object is – but in the physical sciences and engineering, the word length is in some contexts used synonymously with distance. Height is vertical distance; width (or breadth...
0 (zero) or nought is both a number and a numeral. ...
Mathematically, take
if the rotation is around the x-axis (horizontal axis of revolution), or Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...
if the rotation is around the y-axis (vertical axis of revolution). Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...
So here the function p(.) is the distance from the axis and h(.) is generally the function being rotated. The values for a and b are the limits of integration, the starting and stopping points of the rotated shape (i.e. the points delimiting the section of the graph we use).
See also
|
Feeds |
Contact