Area under the curve and its submission.

1. Feb 28, 2009

dE_logics

Within specified limits, the area under the curve of an equation and the submission of all values on that curve will be different right?...just confirming.

2. Mar 1, 2009

maze

What do you mean when you say "submission of all values"?

3. Mar 1, 2009

dE_logics

Submission of all values of f(x) at infinitely small intervals.

4. Mar 1, 2009

5. Mar 1, 2009

dE_logics

Nope that doesn't form any analogy, or at least explicitly state the ambiguity.

What I mean to say here is that there are 2 things that can be computed as a submission of 'something' in the function.

This 'something' might be infinitely small area components determined by dx, and it might be the submission of all the values of f(x) made at infinity small intervals.

I've read many relating this submission of area to the submission of f(x), but what I think, this is not possible.
So I wanna confirm this.

6. Mar 1, 2009

Gib Z

You know, I think you mean summation ! For the original question - as there are an infinite number of points of the curve, the summation of their values will diverge.

7. Mar 2, 2009

dE_logics

Re: Area under the curve and its summation.

The summation will different from the area you mean to say right?

8. Mar 2, 2009

HallsofIvy

What do you mean by the sum of an infinite collection of numbers?

9. Mar 3, 2009

dE_logics

Line integral?:tongue2:

Its not an infinite collection of numbers...its a...sum of numbers in some sorta progression.

10. Mar 3, 2009

HallsofIvy

Then what are you talking about? What "summation of all values on a curve" are you associating with an integral?

11. Mar 3, 2009

dE_logics

Yes of course but normal integral...not line.

I mean...sum of all f(x) within a defined limit, for instance between a to b.

12. Mar 3, 2009

maze

No one here knows what you mean by the "sum of all f(x)". I'm guessing that you are thinking along the lines of the riemann sum, but it is not clear. You are going to have to explain exactly what you mean by "summing all values" for us to be able to help you.

Give a concrete example of what you mean for a specific function. Say, integrating f(x)=x from 0 to 1.

13. Mar 4, 2009

dE_logics

aaa...thanks for notifying me about that.

Take a function y = x (simplest possible)

Suppose I want summation of f(x) (or y) from limits 5 - 11 on x.

So this will be 5+5.1+5.2+5.3.......................10.5+10.6+10.7+10.8+10.9+11

This wont give an accurate result...the intervals at which the addition should occur should be infinity small to give 100% accuracy.

So what I mean to say is when we integrate a function, this summation is not the result right? The summation will be the result in cause of line integral right?

14. Mar 4, 2009

dE_logics

So its not a Riemann sum.

15. Mar 4, 2009

maze

That summation is not the result for anything. Not regular integrals or line integrals or anything. If the spacing gets smaller and smaller, the sum will just get bigger and bigger unbounded, so in the limit as the spacing becomes small, the sum becomes infinity!

However, the sum is very very close to the reimann integral. In fact, all you have to do to make your sum into a riemann sum is to divide by the spacing. eg:
(5+5.1+5.2+5.3.......................10.5+10.6+10.7 +10.8+10.9+11)/(0.1)

In this way, even as the sum becomes infinitely big, the denominator becomes infinitely small in just the right proportions such that the riemann sums converge to the integral.

16. Mar 4, 2009

HallsofIvy

No, it wouldn't. That is the sum of x-values, not f(x) and it is done with a "step" of 0.1. You had said nothing about a particular step interval before.

The Riemann sum, of f(x) with a step of 0.1, between 5 and 11 would be (f(5)+ f(5.1)+ ...+ f(10.9))(0.1). Is that what you are asking about?

17. Mar 4, 2009

maze

in my above post there is an error. It should be multiplying by the spacing as Halls says ,not dividing by it, of course.

18. Mar 5, 2009

dE_logics

Ok I got the answer...thanks a lot everyone!