Introduction
Integrals appear in many practical situations. Consider a swimming pool. If it is rectangular, then from its length, width, and depth we can easily determine the volume of water it can contain (to fill it), the area of its surface (to cover it), and the length of its edge (to rope it). But if it is oval with a rounded bottom, all of these quantities call for integrals. Practical approximations may suffice for such trivial examples, but precision engineering (of any discipline) requires exact and rigorous values for these elements.
Approximations to integral of √x from 0 to 1, with ■ 5 right samples (above) and ■ 12 left samples (below)To start off, consider the curve y = f(x) between x = 0 and x = 1, with f(x) = √x. We ask:
- What is the area under the function f, in the interval from 0 to 1?
and call this (yet unknown) area the integral of f. The notation for this integral will be
As a first approximation, look at the unit square given by the sides x = 0 to x = 1 and y = f = 0 and y = f(1) = 1. Its area is exactly 1. As it is, the true value of the integral must be somewhat less. Decreasing the width of the approximation rectangles shall give a better result; so cross the interval in five steps, using the approximation points 0, 1⁄5, 2⁄5, and so on to 1. Fit a box for each step using the right end height of each curve piece, thus √1⁄5, √2⁄5, and so on to √1 = 1. Summing the areas of these rectangles, we get a better approximation for the sought integral, namely
Notice that we are taking a sum of finitely many function values of f, multiplied with the differences of two subsequent approximation points. We can easily see that the approximation is still too large. Using more steps produces a closer approximation, but will never be exact: replacing the 5 subintervals by twelve as depicted, we will get an approximate value for the area of 0.6203, which is too small. The key idea is the transition from adding finitely many differences of approximation points multiplied by their respective function values to using infinitely fine, or infinitesimal Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite steps.
As for the actual calculation of integrals, the fundamental theorem of calculus The fundamental theorem of calculus specifies the relationship between the two central operations of calculus: differentiation and integration, due to Newton and Leibniz, is the fundamental link between the operations of differentiating In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point. For example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity (respectively, and integrating. Applied to the square root curve, f(x) = x1/2, it says to look at the antiderivative In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i.e., F ′ = f. The process of solving for antiderivatives is antidifferentiation . Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over F(x) = 2⁄3x3/2, and simply take F(1) − F, where 0 and 1 are the boundaries of the interval In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers x satisfying is an interval which contains 0 and 1, as well as all numbers between them. Other examples of intervals are the set of all real numbers , the [0,1]. (This is a case of a general rule, that for f(x) = xq, with q ≠ −1, the related function, the so-called antiderivative In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i.e., F ′ = f. The process of solving for antiderivatives is antidifferentiation . Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over is F(x) = (xq+1)/(q + 1).) So the exact value of the area under the curve is computed formally as
The notation
conceives the integral as a weighted sum, denoted by the elongated s, of function values, f(x), multiplied by infinitesimal step widths, the so-called differentials, denoted by dx. The multiplication sign is usually omitted.
Historically, after the failure of early efforts to rigorously interpret infinitesimals, Riemann formally defined integrals as a limit In mathematics, the concept of a "limit" is used to describe the behavior of a function as its argument or input either "gets close" to some point, or as the argument becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely. Limits are used in calculus and other branches of of weighted sums, so that the dx suggested the limit of a difference (namely, the interval width). Shortcomings of Riemann's dependence on intervals and continuity motivated newer definitions, especially the Lebesgue integral In mathematics, Lebesgue integration refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a sub-domain of the real line or a higher dimensional Euclidean space with respect to the Lebesgue measure. This article focuses on the more general, which is founded on an ability to extend the idea of "measure" in much more flexible ways. Thus the notation
refers to a weighted sum in which the function values are partitioned, with μ measuring the weight to be assigned to each value. Here A denotes the region of integration.
Differential geometry Differential geometry is a mathematical discipline that uses the methods of differential and integral calculus, as well as linear and multilinear algebra, to study problems in geometry. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for its initial development in the eighteenth and, with its "calculus on manifolds In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold. Thus a line and a circle are one-dimensional manifolds, a plane and the surface of a ball are two-dimensional manifolds,", gives the familiar notation yet another interpretation. Now f(x) and dx become a differential form In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. A differential form of degree k, or k-form, on a smooth manifold M is a smooth section of the kth exterior power of the cotangent bundle of M. The set of all k-forms on M is a, ω = f(x) dx, a new differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another d, known as the exterior derivative In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a form of degree one, to differential forms of higher degree. Its current form was invented by Élie Cartan appears, and the fundamental theorem becomes the more general Stokes' theorem In differential geometry, Stokes' theorem is a statement about the integration of differential forms which generalizes several theorems from vector calculus. William Thomson first discovered the result and communicated it to George Stokes in July 1850. Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result,
from which Green's theorem In physics and mathematics, Green's theorem 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. It is the two-dimensional special case of the more general Stokes' theorem, and is named after British mathematician George Green, the divergence theorem In vector calculus, the divergence theorem, also known as Gauss’s theorem , Ostrogradsky’s theorem (Mikhail Vasilievich Ostrogradsky), or Gauss–Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface, and the fundamental theorem of calculus The fundamental theorem of calculus specifies the relationship between the two central operations of calculus: differentiation and integration follow.
More recently, infinitesimals have reappeared with rigor, through modern innovations such as non-standard analysis Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number. Not only do these methods vindicate the intuitions of the pioneers, they also lead to new mathematics.
Although there are differences between these conceptions of integral, there is considerable overlap. Thus, the area of the surface of the oval swimming pool can be handled as a geometric ellipse, a sum of infinitesimals, a Riemann integral, a Lebesgue integral, or as a manifold with a differential form. The calculated result will be the same for all.
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