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Videos of integrals

Integrals

Integrals

Integration is a core concept of advanced mathematics, specifically in the fields of calculus and mathematical analysis. Given a function f(x) of a real variable x and an interval [a,b] of the real line, the integral

is equal to the area of a region in the xy-plane bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b, with areas below the x-axis being subtracted.
The term "integral" may also refer to the notion of antiderivative, a function F whose derivative is the given function f. In this case it is called an indefinite integral, while the integrals discussed in this article are termed definite integrals. Some authors maintain a distinction between antiderivatives and indefinite integrals.
The principles of integration were formulated by Isaac Newton and Gottfried Leibniz in the late seventeenth century. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if f is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of f is known, the definite integral of f over that interval is given by:

Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. A rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the nineteenth century, more sophisticated notions of integral began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. A line integral is defined for functions of two or three variables, and the interval of integration [a,b] is replaced by a certain curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space. Integrals of differential forms play a fundamental role in modern differential geometry. These generalizations of integral first arose from the needs of physics, and they play an important role in the formulation of many physical laws, notably those of electrodynamics. Modern concepts of integration are based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue.

Definition of calculus

Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. Most basically, calculus is the study of change, in the same way that geometry is the study of space.
Calculus has widespread applications in science and engineering and is used to solve problems for which algebra alone is insufficient. Calculus builds on algebra, trigonometry, and analytic geometry and includes two major branches, differential calculus and integral calculus, that are related by the fundamental theorem of calculus. In more advanced mathematics, calculus is usually called analysis and is defined as the study of functions.

The chain rule and compound functions

Compound functions:
A compound function is a function that is formed of two or more functions, where one depends on the other, behaving like a chain where each link is a function that is within another one and this is contained as well in another one and so on, to derive these compound functions we began from the most internal function to most external, considering that the following one always is going to depend on the previous one.
to derive the compound functions we must start by deriving the the most internal function until the most external , or by the simplest one until the most complex.

Properties of the limits

Limits

Definition of limit :

In mathematics, the concept of limit is used to describe the tendency of a succession or a function. The idea is that in a succession or a function, when speaking of limit, we say that has one if it is possible to be approached a certain number as much (that is, the limit) as we want

The study of the limits of functions of several variables is much more complex that the ones of of one variable , because in this, it only has two ways to approach a point, by the right or the left; whereas in the case of several variables a infinity of ways exists to approach us a point, as it shows in the figure 1.

figure 1

theorems of limits:

Theorem of limit1:

If k is a constant and to any number , then:

Theorem of limit2 :

To any number "a":

Theorem of limit3:

If m and b are two constants then:

Theorem of limit4:

Theorem of limit5:

Theorem of limit6:

If "f "is a polynomial and "a" is a real number, then

Theorem of limit7:

If q is a rational function and to it belongs to the dominion of q, then:

Theorem of limit8:

Solved Ejercises :

Solve the following limits indicating the property or properties that are applied in each ste

According to the theorem of limit 1:

Lim 77=77

x---3

f(x)= 3x-7 = mx+b that's why we apply the theorem of limit 3:

Lim 3x+7 = 3(5)-7 = 15-7

x--5

Lim = 8

x---5

Indeterminations

there are several types of indeterminations, among them: [refers to the limit to infinite and 0 the limit to 0 (not to number 0)]