can anyone tell what is integration?

Question

can anyone tell what is integration?

Answer 1

Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral



\int_a^b \! f(x)\,dx \,



is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of ƒ, 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 ƒ. 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 independently by Isaac Newton and Gottfried Leibniz in the late 17th century. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if ƒ is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of ƒ is known, the definite integral of ƒ over that interval is given by



\int_a^b \! f(x)\,dx = F(b) - F(a)\,



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 integrals 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. There are many modern concepts of integration, among these, the most common is based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue.

Answer 2

in short integration is opposite of differentiation.

Answer 3

E.J. In my layman's terrms integrationis the slicing up of a function into thin slices and then adding them up. It will give one the area under the curve on a graph . Lets take the function 2pi R= circumference of a circle We wish to find the area of the circle by integration. Lets make the circumference :2Pi R the base of an asoseles triangle and the height R . The area of the triangle is :(2pir x R) /2=Pi R ^2 Lets now take our triangle and start dividing it into many isoceles triangles each with the same height. Regardless of how many little triangles we make we can take the area

of each and add them up and we will have the area of the original triangle. Now, we take the little triangles and put their apexes at a common point like the center of a circle. As we put them around the center point the triangles begin to resemble somewhat of a circle with many clipes in its circumference( the bases of the little triangles) Remembering that a line has no width, we see as we divide the triangles more our rough circle looks more like a truer circle( the clipes get smaller) We say to ourselves that since a line has no width we can make the triangle bases as narrow as we wish Ok: we make each base so small and so close to zero that the hypotenuse (R of a circle) is so close to the height of the triangles that the difference is only .000000000000000ad infinitum 1 . or essentially zero which makes the height of the triangles and R (radius of circle) the same to a degee at least as accurate as pi. Since we know that the sum of all the bases of the little triangles is 2piR and their heghts are now essentially equal to R ( their hypotenuses) and we have made a circular area of them which we call a circle( a figure formed by points that are the bases of an infinite number of isoceles triangles having their apexes at a common point)

To know the area we simply integrate( Add up all their base=2piR multiply by their height and divide all by 2 to find the area of the circle=( 2piR x R)/2 =piR^2 So we have now integrayted 2pi R into Pi R ^2. A short way would have been to simply take 2piR make R > R^2 and put the value of the exponent 2 below as a divisor=( 2piR^2)/2 = pi R^2. Remember": Integration does not give the exact value of anything as long as pi is involved be cause pi is no exact( only a close- close approximation) We can contiue this to find the surface area of a sphere ( the bases of an infinite number of idential pyramids each having their apexes at the center of the sphere and knowing the total of their bases = the surface area of the sphere and the volime of a sphere is 1/3(R of sphere)x sum of bases which is 4pir^2 which integrates into 4piR^2 x1/3R=4/3piR3, but first we have to find the surface area of a sphere, We do this the same way we would take an orange and cut an infinitly thin slice across its middle, then rotate the orange an amount equal to the first slice and make another infinitly small slice etc. After we have rotated the orange piR amount we have sliced up the whole orange. Enlarged, each piece would look like a wedge( a triangular look for their surface area; however, remembering a line has no width the slices take on the resemblences of just a line. We stack all the slices up and knowing we rotated the orange only 1/2 of a revolution to make all the slice(2R) , we have a stack 2R in height each slice having a circunference of 2piR ie we have a cylinde 2R in height and 2piR in circumference . The vertical surface area of the cylinder is 2piR x 2R=4piR^2. Which we integrate to 4/3piR^3 to get the volume Volume = Bases of infinite number of identical trianglles (4piR^2 x 1/3 R (height of pyramids-= 4pi R^2 x 1/3 R =4/3 piR^3. Hope this helps. Good Luck

Answer 4

in an x-y plane, for a mathematical function when applied, the union of all resultant values under x and y is called integration value of tat function, under that plane.

Answer 5

well, putting the mathematical theorem would be a tad difficult, but I can tell you the gist of the theorem.

The Fundamental Theorems of Calculus dictates the use of an integral. It states that if f is continuous on a closed interval [a, b] and a function F'=f, then the integral of F' from that closed interval is equal to f(b)-f(a). It states that a function can be represented by infinitely many anti-derivatives (integrals).

What the integral of a function represents is the area under the curve that function makes when plotted on a graph. Of course, this is about as boiled down as it gets. There is significantly more to the entire answer than what I just gave you.

Answer 6

Draw an equation on squared graph paper. and take two limits like x=2 to x=7 and draw them.

then shade the area enclosed by the equation, x=2, x=7 and the x-axis. the area of the shaded region is given by the integration of the equation with limits 2 to 7.

Answer 7

integration means area under any curve

Answer 8

It means that you have to find the definite integral (as opposed to the antiderivative).



http://en.wikipedia.org/wiki/Integral

Answer 9

The opposite of differentiation

Answer 10

The Integration means the summation of all the ordinates of the curve between given range of a curve which is multiplied by width between two ordinates which tends to zero. Say y=f(x)

I=Lt dx->o §(y1+y2+.........+yn)dx

I=§ y.dx which is nothing but area of the curve between given range