Re: [math-fun] calculus question
Andy is certainly right that a definition of integral that uses equal-with intervals and, say, always left endpoints (EWLE) is easier to conceive of than the more general "any partition whose mesh approaches 0, and any points within the intervals" (APAP) definition. Pedagogically, there's no question that I'd introduce integrals, and confine all or almost all examples to the EWLE (or EWRE) type. But when first learning calculus I recall being a bit confused by which endpoint -- or was it the midpoint -- to evaluate the function at. When later (I think when using the ubiquitous text by G.B. Thomas) I learned the APAP definition, it was a relief not to have to be concerned about such details as the exact partition and where to evaluate the function. I would *state* but certainly not *prove* such a theorem in an intro course (other than to honors math majors, who might use a book like Apostol's). Jim did, by the way, say this is an honors course. (Is it honors for anyone, or mainly honors math majors?) ------------------------------------ Pedagogy aside, certainly one could cook up a function having the value 1 on all points of the form k/n for integers k,n such that 0 <= k/n <= 1, and the value 2 at other points. But this doesn't answer the interesting math question brought up by Jim: << Stewart says that a function f on the interval [a,b] is integrable with definite integral I if for all epsilon > 0 there exists a delta > 0 such that EVERY Riemann sum for f(x) on [a,b] whose mesh (the maximum of the widths of the sub-intervals) is less than delta has value within epsilon of I.
[what I called APAP] << The student asked, would it be equivalent to require the sub-intervals to be of equal width (still allowing the sample-points to be arbitrary within the sub-intervals)?
Let's call this method EWAP. So is EWAP integrable equivalent to APAP integrable, and yielding the same values? Obviously APAP implies EWAP since equal width is a special case of arbitrary partitions. Conversely, there is a chance that a strange singularity can be created near 0, say, of a function f:[0,1} -> R, such that f is EWAP integrable, but not APAP integrable. Good question. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
On Feb 6, 2008 4:13 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Andy is certainly right that a definition of integral that uses equal-with intervals and, say, always left endpoints (EWLE) is easier to conceive of than the more general "any partition whose mesh approaches 0, and any points within the intervals" (APAP) definition.
Pedagogy aside, certainly one could cook up a function having the value 1 on all points of the form k/n for integers k,n such that 0 <= k/n <= 1, and the value 2 at other points.
Of course. I don't know what I was thinking when I suggested otherwise. Any integral scheme that only looks at countably many points will be defined more often than the Riemann interval, since you can just fudge those points to make the integral work on an otherwise not-integrable function.
But this doesn't answer the interesting math question brought up by Jim:
<< Stewart says that a function f on the interval [a,b] is integrable with definite integral I if for all epsilon > 0 there exists a delta > 0 such that EVERY Riemann sum for f(x) on [a,b] whose mesh (the maximum of the widths of the sub-intervals) is less than delta has value within epsilon of I.
[what I called APAP]
<< The student asked, would it be equivalent to require the sub-intervals to be of equal width (still allowing the sample-points to be arbitrary within the sub-intervals)?
Let's call this method EWAP. So is EWAP integrable equivalent to APAP integrable, and yielding the same values?
I don't have a full proof yet, but I'm pretty sure the answer is yes.
Conversely, there is a chance that a strange singularity can be created near 0, say, of a function f:[0,1} -> R, such that f is EWAP integrable, but not APAP integrable.
The thing is, once the mesh size is less than 10^-100, the effect of the singularity at 0 is tiny, so the AWAP integral will still converge whenever the EWAP one does. In the quest for the most concrete definition of an integral that exists exactly when the Riemann integral exists, EWLE goes too far, and ends up being defined too often. But I think that is only because it only samples finitely many points. Is it possible to eliminate the confusing "choose any point in the interval" part of the definition, while still producing an integral that converges in exactly the same situations as the Riemann integral? For example, suppose we let the fixed size be any value, rather than only 1/n (allowing the last partition to be a different size) by defining a new integral, EWLE' = lim {e->0} sum_{i=0}^{[1/e]}ef(ie) Are there any functions where APAP does not converge, but EWLE' does? I'm less sure of myself here than in the EWAP case, but I suspect that EWLE' only converges when APAP converges
Good question.
--Dan
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
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Dan Asimov