dx meaning in integration????

Hi. I am confused why an integrated equation is called dx.

Could you explain a bit more!

Could you explain a bit more!

...

I'll try to write something up later today, and edit this post accordingly. It might take several hours though.

EDIT: Ok, here it is!

Consider a function $f: \mathbb{R} \rightarrow \mathbb{R}$. A function is like a 'number machine' - you put a real number in, and it converts it into another real number (possibly the same one). For example, we could have $f(x)=x^2$, which sends a real number x to its square. Notice that the variable x is not fundamental, it's merely the name we've chosen to give the input variable. For example, if g sends y to y^2, and g and f have the same domain, then f and g are identical functions.

Given our function f, let F (note this F is capital) be given by

$[br]F(p)= \int_a^p f(x) dx[br]$

Here, a is a fixed point, and p is a variable point. Convince yourself that F really is a function of p - that is, if you specify p, then the value of F(p) is uniquely determined. In particular, F(p) is equal to the area of the region shaded yellow in this picture:



again, we integrated with respect to x, but this was not fundamental, we could have equally written $F(p) = \int_a^p f(y) dy$. However, the P is fundamental, since changing the location of the point P on the horizontal axis will change the area of the yellow region.

Now, what is the derivative of F? Well, the derivative is equal to $\frac{F(p+h)-F(p)}{h}$ when h is small.
This is equal to the area of the green region, divided by h. However, the green region is approximately equal to a rectangle of width h and height $f(p)$, as shown in the diagram. [Caveat: Actually, I have drawn the wrong rectangle in this diagram - it should be a little shorter, so it has height $f(p)$ rather than $f(p+h)$. So the derivative of F with respect to p is F(p) as required.

I'm not really sure if this answers OP's question or not, but it difficult to answer it unless OP has a definition of integration. I have tried to give this (informally) here; hopefully it helps.

Uhmmm I'm not sure I understand what you mean. The integration equation is not called dx, it's the EQUATION. dx is a part of the equation. About the equation of a line, it is y in term of x, or y with respect to x. So integrating dx make perfect sense. And the final equation doesn't contain dx (it is ax^n+1 or something) so I'm not sure what you mean by "this is still called dx".

As you move on, you will know that you can integrate dy, dx, dt or whatever d you can come up with. Integrating is just "finding the area under a line".

‘dx’ means that you integrate with regards to x.

dx in this case means integrate in terms of x.

Thank you for your very in-depth response.

So as I understand it you are confused between two separate 'definitions'
(i) dx means a small piece of x
(ii) dx means integrate with respect to x

Perhaps the following remarks might help.

(i) Don't think of dx as having any meaning by itself. It is the dx, together with an integral sign, that has meaning. For example, the integral $\int x^2$ has no meaning, because we haven't said what variable we are integrating with respect to, so we need the integral sign together with a 'dx' or 'dy' for our notation to make sense.

(ii) But where does the 'dx' notation come from? And what is the connection with 'a small piece of x'? This is easily explained if you know the definition of the definite integral. To find, 'the area under a curve', you approximate this area with a bunch of rectangles. Usually your curve is of the form y=f(x), and to get a good approximation we use a lot of tall, thin rectangles. 'Thin' means that their width - measured along the x axis, is small. So dx reminds us that our integral is defined by using lots of rectangles which are small in the x direction.

You can read more about (ii) by looking here, although it is a little technical:
http://math.feld.cvut.cz/mt/txtd/1/txe3da1a.htm

EDIT: You said you define integration to be the inverse of differentiation. This is a good definition for the indefinite integral. The definition of the definite integral is that it is the area under the curve between the limits. It is a theorem (i.e. a non-obvious fact) that these definitions amount to the same thing - I tried to explain why in my earlier post.

Okay, thanks. This has made dx clearer to me. But you only talked about integration for finding the area underneath a curve; yet does the concept of the rectangles work when integrating the dy/dx of a line to get its original equation?

Yes it does - I don't think I can explain this any better than I already have; perhaps you should reread the paragraph marked 'EDIT' in my previous post, and also my post before that, where I show (or, at least, give a plausible argument that) the 'inverse of differentiation' concept is consistent with the 'area under the curve' concept.

It might help to realise that the indefinite integral $\int f(x) dx = F(x) + C$ is the same thing as $\int_a^x f(y) dy = F(x) - F(a)$. In the first expression, the arbitrary constant is +C and in the second it is -F(a), but since these constants are arbitrary, these amount to the same thing.