Integral Notation - Missing Integrands

Date: 07/17/2003 at 21:06:33
From: Chris
Subject: Integral Notation - "Missing" Integrands

Hi,

I have seen some integral notation used that I am not familiar with. 
It looks like this:

 /
 | dx f(x) + ...
/

There does not seem to be an integrand (i.e. a function being 
integrated). I'm not sure if f(x) is to be integrated. I have two 
theories, but I can't see the point in writing the expression as it is 
if either of my theories is correct.

My theories about what this might mean:

1) The above notation is the same as writing:

 /
 | 1 dx f(x) + ... (note the explicit 1 here)
/

=

(x + C) * f(x) + ... (where C is a constant of integration)

2)  The rest of the expression is to be integrated with respect to x.

If (1) is correct, then what was the point of writing the integral - 
why wasn't (x + C) just written instead? If (2) is correct, then how 
does one know when to "stop integrating" (i.e. if there is some term 
to be added on to the expression that is not to be integrated, how is 
it distinguished?).

I have seen this recently in multi-variate calculus, i.e. when x is in 
R^n rather than R: does this situation justify the use of the integral 
notation somehow?

Thanks in advance.

Date: 07/18/2003 at 12:43:05
From: Doctor Peterson
Subject: Re: Integral Notation - "Missing" Integrands

Hi, Chris.

It is common to learn about integration in such a way that the "dx" 
seems to be a marker for the end of the integral, as if the "long S" 
were a left parenthesis and the "dx" were the right parenthesis. But 
it doesn't work that way. In fact, what you are integrating is the 
product of a function and dx; and multiplication is commutative! So 
these mean the same thing:

   /                 /
   | f(x) dx   and   | dx f(x)
  /                 /

If you then add something, you must use parentheses if it is to be 
part of the integral:

   /                    /
   | dx f(x) + g(x) = [ | f(x) dx] + g(x)
  /                    /

is the sum of an integral and a function, while

   /                      /
   | dx (f(x) + g(x)) =  | (f(x) + g(x)) dx
  /                      /

is the integral of the sum of two functions.

That is, presumably the integral has higher precedence than addition, 
so you "stop integrating" at the first plus sign. But even then, I'm 
not positive that this rule I just made up is always followed; let me 
know if you think it doesn't fit the practice in your text, and show 
me an example.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/19/2003 at 07:11:11
From: Chris
Subject: Thank you (Integral Notation - "Missing" Integrands)

Doctor Peterson,

Thank you for your quick and helpful reply.

I was indeed taught that integration begins with the "long S" and ends 
with the (for example) dx.

I have, however, seen the following notation:

 /
 |      dx
 | ------------
 |  f(x) + g(x)
/

and assumed it was a convenient notation rather than being a 
justifiable mathematical expression.

Perhaps I need to go and look at calculus from first principles again 
to see why this is the case.

Thank you again.

Chris