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
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