what is itegration?

[computerages]

hello everyone!

 

I am having really a hard time "understanding" what do we mean when we say 'integrate a function'.. I know it's a backward derivative, anti derivative, but I don't understand "how" it is..

 

for example, i know and I "understand" that derivate of a function is the rate at which that function changes, and it's calculated by taking a limit of zero to a specific point...but integration is really burning my neurons...

 

also, do u know any good website which shows analytical approach to solve for definite and indefinite integrals.?? or if possible, could someone explain?

[ajb]

Thinking of definite integrals separately from derivaties, the integral is the "area under the curve". In essence this was Riemann's definition.

 

At first glance it is not obvious that intergration and derivatives have anything to do with each other. It is a deep fact from the so called "fundamental theorem of calculus" that they are "inverses" of each other.

 

To answer you question you have to look at this theorem. See for example

 

http://mathworld.wolfram.com/Integral.html

http://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html

[the tree]

At first glance it is not obvious that intergration and derivatives have anything to do with each other.

I think it is, maybe it's the way I was taught that I was introduced to integration firstly in its indefinite sense to underive stuff. And then definite integrals as a brilliantly useful coincidence.

[Bignose]

It is probably impossible to stress how important the Fundamental Theorems of Calculus really are. Well, hence the word 'Fundamental'. See if ajb's links help your question, or if any of the other first links that come up from searching "fundamental theorems of calculus" in google. There are lots of different way of explaining it on the just the first page of Google results -- one of them will probably jive. In fact, if one of them helps a lot, come back and tell us which one was best for you. I like the wikipedia entry best myself -- the animation of the converging Riemann sums was good as was the more than one proof -- but I'd like to know which one worked best for you and why.

[EvoN1020v]

Integration can also find you the "original" function.

 

For example:

[math]f(x)=x^2[/math] is the orginial function.

First derivative is: [math]2x[/math].

 

Suppose you have the function of [math]2x[/math] and you want the integration of it, you use the Integration Rule which is [math]\frac{x^{n+1}}{n+1}[/math].

 

So you get: [math]\frac{2x^{2}}{2}[/math]. The "2" will cancel each other so you have the original function: [math]x^2[/math].

 

Integration is just a kind of math that gives you the "orginial" function of the function that you have in firsthand.

[timo]

It is probably impossible to stress how important the Fundamental Theorems of Calculus really are.

I think it´s only one that´s called fundamental theorem of calculus, the one ajb mentioned.

[Bignose]

I think it´s only one that´s called fundamental theorem of calculus, the one ajb mentioned.

 

Not to nitpick, but Mathworld calls them the Fundamental Theorems.

 

But whether it is one theorem or two theorems does not really matter, the relationship between the integral and derivative is an extraordinarily important part of calculus and if the OP has any further question, they should be sure to ask them so that the important of this concept can be conveyed appropriately.

[Ragib]

I believe it is Fundamental TheroremS, i've seen it in multiple places, and there are a few variations, small differences but lead to slightly different results..