Can I know the logic behind the integration&differentiation?

Question

Clarify me about the difference between sigma,integration&addition.

Answer 1

Hey there!

The sigma notation is commonly used to represent the sum of the series. The series can either be arithmetic, geometric or infinite.

Integration is normally used to find the area under the curve. When you're more knowledgable in calculus, you can use integration to find surface areas and volumes of unique solids. Integration can also help you to find arc lengths of the curves and also helps you prove the circumference of a circle.

Derivatives are commonly used in calculus, which helps you in finding the slope of a curve. Derivatives can also be useful in solving limits, where some of the limits lead to indeterminate forms i.e. 0/0.

Addition is simply the sum of numbers or variables.

Hope it helps!

Answer 2

Addition is a one time operation performed on a set of specific given terms that may be related or not.
Sigma is repeated addition performed on a set of related number solutions, usually within a specific boundary (Sigma is Greek for capitol S and represents a SUM of the solutions to whatever given equation follows that symbol) for a series of integers plugged into the equation.
While the integral sign (another capitol S representing summation. The 'S' used in Newtons time) represents the sum of solutions of the dependant variable for all values of the independent variable plugged into an equation. A better way to understand it is to think of it as the area under a curve bounded by the integration limits. This is a better summation process than sigma because it takes into account all values of the independent variable not just the integers so it is a result of a more smooth and continuous operation.

Answer 3

sigma is a summation symbol that represents the sum of a collection or series. It is related to integration in that integration, when looking for area, looks at the sum of the areas of individual rectangles (or trapezoids, that exist between a curve and an axis or between two curves.

Differentiation and integration can be looked at as inverses. The purpose of differentiation is to look at the simultaneous change in the function value (y) as x acheives a given value. Integration, instead, can be used to describe a "family" of functions possessing the given change (slope) at the given x values. So they have reversed outcomes. Integration actually has many different applications in mathematics.

Answer 4

sigma is used to represent a sum of a series for example:

sigma(x^3, 1, 5)

Sum of the series X^3 from 1 to 5
1^3 + 2^3 + 3^3 + 4^3 + 5^3

integration USUALLY solves for area within a curve. This is also called an anti-derivative

derivatives usually gives you the instantaneous rate of change of a function

Answer 5

Sigma is used to sum a series. If you have a series s_n = {x^n} then you can write its sum as

sigma(s_n,n=0,N) = x^0 + x^1 + x^2 + ... + x^N

if you let N=infinity this is called a power series.

Integration is used to find the area under the curve, and involves infinite sums. You can approximate the area under a curve by using a trapezoidal rule by discretizing your x-axis into equally sized pieces of size deltax. then you can approximate the area under the curve over this deltax as deltaA = (y(x) + y(x+deltax))/2*deltax

then the total area from a to b is approximately

sigma(deltaA_n, n=0,n=(b-a)/deltax) so you sum each deltaA between x=a,b. Then integration is the exact area, and is found by taking the limit as deltax goes to zero,

lim(sigma(deltaA_n,n=0,n=(b-a)/deltax),deltax->0). So integration is essentially an infinite sum where n (the number of chunks under the curve) goes to infinity.

Answer 6

so which you assert that because of the fact so a techniques technological know-how hasn't got here upon all of the solutions that the only logical ingredient is to have faith that an all-powerful being could have created each thing, top? it is relatively not greater logical then understanding that we don't understand or understand each thing approximately technological know-how yet, yet that the lack of ability of awareness does no longer imply that there is a God . because of the fact there is plenty that maintains to be unknown, theory in a God or no longer is in user-friendly terms a private theory it is in accordance with ones own existence experience, training, parenting and how their techniques computes that counsel.

Answer 7

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

\int_a^b f(x)dx

represents 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.

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. The fundamental theorem of calculus asserts that an antiderivative can be used to compute the integral over an interval. Some authors maintain a distinction between antiderivatives and indefinite integrals.

The idea of integration was formulated in the late seventeenth century by Isaac Newton and Gottfried Wilhelm Leibniz. Together with the concept of a derivative, integral became a basic tool 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.
ntegrals 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 at first, but eventually we demand exact and rigorous answers to such problems.

Measuring depends on counting, with the unit of measurement as a key link. For example, recall the famous theorem in geometry which says that if a, b, and c are the side lengths of a right triangle, with c the longest, then a2 + b2 = c2. If we take the side of a square as one unit, its diagonal is √2 units long, a measurement which is greater than one and less than two. If we take the side as five units, the diagonal is at least seven units long, but still not perfectly measured. A side of 12 units will have a diagonal of about 17 units, but slightly less. In fact, no choice of unit that exactly counts off the side length can ever give an exact count for the diagonal length. Thus the real number system is born, permitting an exact answer with no trace of approximating counts. It is as if the units are infinitely fine.
Approximations to integral of √x from 0 to 1, with ■ 5 right samples (above) and ■ 12 left samples (below)
Approximations to integral of √x from 0 to 1, with ■ 5 right samples (above) and ■ 12 left samples (below)

So it is with integrals. For example, consider the curve y = f(x) between x = 0 and x = 1, with f(x) = √x. In the simpler case where y = 1, the region under the "curve" would be a unit square, so its area would be exactly 1. As it is, the area must be somewhat less. A smaller "unit of measure" should do better; so cross the interval in five steps, from 0 to 1⁄5, from 1⁄5 to 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. The total area of these boxes is about 0.7497; but we can easily see that this is still too large. Using 12 steps in the same way, but with the left end height of each piece, yields about 0.6203, which is too small. As before, using more steps produces a closer approximation, but will never be exact. Thus the integral is born, permitting an exact answer of 2⁄3, with no trace of approximating sums. It is as if the steps are infinitely fine (infinitesimal).

The most powerful new insight of Newton and Leibniz was that, under suitable conditions, the value of an integral over a region can be determined by looking at the region's boundary alone. For example, the surface area of the pool can be determined by an integral along its edge. This is the essence of the fundamental theorem of calculus. Applied to the square root curve, it says to look at the related function F(x) = 2⁄3√x3, and simply take F(1)−F(0), where 0 and 1 are the boundaries of the interval [0,1]. (This is an example of a general rule, that for f(x) = xq, with q ≠ −1, the related function is F(x) = (xq+1)/(q+1).)

In Leibniz's own notation — so well chosen it is still common today — the meaning of

\int f(x) \, dx \,\!

was conceived as a weighted sum (denoted by the elongated "S"), with function values (such as the heights, y = f(x)) multiplied by infinitesimal step widths (denoted by dx). After the failure of early efforts to rigorously define infinitesimals, Riemann formally defined integrals as a limit of ordinary 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, which is founded on an ability to extend the idea of "measure" in much more flexible ways. Thus the notation

\int_A f(x) \, d\mu \,\!

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, with its "calculus on manifolds", gives the familiar notation yet another interpretation. Now f(x) and dx become a differential form, ω = f(x)dx, a new differential operator d, known as the exterior derivative appears, and the fundamental theorem becomes the more general Stokes' theorem,

\int_{A} \bold{d} \omega = \int_{\part A} \omega , \,\!

from which Green's Theorem, the divergence theorem, and the fundamental theorem of calculus follow.

More recently, infinitesimals have reappeared with rigor, through modern innovations such as non-standard analysis. 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, as a sum of infinitesimals, as a Riemann integral, as a Lebesgue integral, or as a manifold with a differential form. The calculated result will be the same for all.

summation::
Summation is the addition of a set of numbers; the result is their sum. The "numbers" to be summed may be natural numbers, complex numbers, matrices, or still more complicated objects. An infinite sum is a subtle procedure known as a series. Note that the term summation has a special meaning in the context of divergent series related to extrapolation.

differentiation::
In calculus, a branch of mathematics, the derivative is a measurement of how a function changes when the values of its inputs change. The derivative of a function at a chosen input value describes the behavior of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In general, the derivative of a function at a point determines the best linear approximation to the function at that point.[1]

The process of finding a derivative is called differentiation. The fundamental theorem of calculus states that differentiation is the reverse process to integration.

Differentiation has applications to all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. Newton's second law of motion states that the derivative of the momentum of a body equals the force applied to the body. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories. By applying game theory, differentiation can provide best strategies for competing corporations.

Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear throughout mathematics, in fields such as complex analysis, functional analysis, differential geometry, and even abstract algebra.