What is integration?

Question

I have gone through a lot of books on integration,, but still i am not able to grasp it
Please make me understand it in the simplest possible way,,,
Also tell how integration of circumference of a circle is its area

Thanks.........

Answer 1

Simply put, integration is the opposite of the derivative. But to really grasp it in an intuitive way, think of it in terms of velocity. When you drive a car, you start at speed 0 mph, then you go faster and faster until you get to, say 60 mph. Then you have to slow down to 30 mph, but it takes time to get slowed down too. Integration can tell you how far you traveled while speeding up and slowing down.



If you break the problem into small time slices, like a second, it might help. Think first about going 0-60 feet per second in 6 seconds. Let's say you increase by 10fps/sec. So in the first second you go from 0-10fps. In the second you go 10-20fps. In the third you go 20-30 fps, and so on. If you average it out, you are going about 5 fps for the first second, or 5 feet. Then 15 fps for the second second, or 20 total feet. Then 25 fps for the third second, or 45 total feet. By integrating you can get the complete area under the curve of acceleration (total distance traveled), instead of the approximation I'm working toward.

Answer 2

Suppose you have a graph of the speed of a moving car, expressed as f(x), and you want to know how far it goes between, say, time x=0 and time x=1. Over a small interval, say "dx," its speed is approximately f(x). and the distance it travels is f(x) dx. Like, 50 feet per second times 1 second = 50 feet.



What integration does is to add up these small intervals (infinitesimally small), to get the distance. That's why the integral from 0 to 1 of f(x) dx is the distance traveled during that time interval.



So where does the antiderivative come in? If you look at the rate of change of distance, that's the speed. If the car went from one point to another point 50 feet away in 1 second, it's speed is 50fps. That's why the area under the curve represents the change in value of the antiderivative of the curve.



I hope that helps.

Answer 3

Integration (from the Latin integer, meaning whole or entire) generally means combining parts so that they work together or form a whole. In information technology, there are several common usages:



1) Integration during product development is a process in which separately produced components or subsystems are combined and problems in their interactions are addressed.



2) Integration is an activity by companies that specialize in bringing different manufacturers' products together into a smoothly working system.



3) In marketing usage, products or components said to be integrated appear







to meet one or more of the following conditions:



A) They share a common purpose or set of objectives. (This is the loosest form of integration.)



B) They all observe the same standard or set of standard protocol or they share a mediating capability, such the Object Request Broker (ORB) in the Common Object Request Broker Architecture (CORBA).



C) They were all designed together at the same time with a unifying purpose and/or architecture. (They may be sold as piece-parts but they were designed with the same larger objectives and/or architecture.)



D) They share some of the same programming code.



E) They share some special knowledge of code (such as a lower-level program interface) that may or may not be publicly available. (If not publicly available, companies have been known to sue to make it available in order to make competition fair.)

Answer 4

Integration via areas. The function is a made from 2 applications: (x) * (cosx) call one among them "u" and the different "dv" pick the "u" so as that its spinoff simplifies u = x du = a million*dx dv = cosx v = sinx that's for the formulation setup, that's u*v - essential (vdu) =xsinx - essential(sinx * a million * dx) =xsinx - (-cosx) + C = xsinx + cosx + C

Answer 5

integration gives you the area under a curve.



If you take a width dx under a line f(x). then the area of the strip under curve is f(x)dx.

Now if you take the area of all the strips under a graph with a certain limit, Then ∑f(x)dx.

Therefore, the get the exact area under the curve you let dx ---> 0, hence area = ∫f(x) dx.