(1) My response is in the forum. Not sure why it ws not visible to you, might be a glitch on one of our servers.

(2) The only behavior from that NIntegrate example I can elicit is where it gives the error message and returns unevaluated. I find it to be a very useful error message, by the way. Returning unevaluated with no indication as to why would be unhelpful, not to mention annoying.

(3) The warning messages associated with precision could be better. The usual reason such a message appears is that one requests WorkingPrecision that is higher than the precision of the input. So again, the message is useful, but sometimes people fail to understand what it is telling them and that could perhaps be due to the wording.

(4) The questions posed do not have inputs where I can actually figure out what is the underlying issue.

Daniel, your reply did not post in the User Group. I got your message and logged in on the User Group web site but your reply is not here. It seems you duplicated my issue, which is why I did not attach a worksheet, since I wanted an independent verification outside of my Mathematica application.

The issue is to me clear: somehow the simple numerical integration does not execute on the first attempt to evaluate but it does execute on the second attempt with the message you posted. Why does it execute the second time and throw out an inane comment? If I use the same process on a more complete numerical integration with the precision and accuracy goal and the working precision indicated, the same thing happens: it does not execute on the first iteration but does on the second, even though all parameters are not defined and so the numerical integration cannot proceed...yet on the second evalution is does with the second inane comment I posted.

Why does this happen and how do I keep it from happening? The reason I would like some clarity is that these integrals are defined as functions that are embedded in more complex expressions and I cannot assure that the second and unexpected execution has not created an expression that is incorrect in the more complete expression.

Now, my concern may not mean anything to you but to me it means that I cannot trust the results that I calculate. The reason is that I am preparing a worksheet to post in the physics section that identifies a particular integral that needs to be numerically evaluated and there are some issues I am addressing on how to interpret the results given how Mathematical performs the numerical calculations.

I don't understand what the expected output is for the example. You have variables x and a but you specify limits only for a, Second, what is the last parameter for? Usually, limits of integration are, for example, like {a, 1., 2.} -- the extra value doesn't seem to have any use -- and it is not included in the documentation.

The error message is clear. Since you did not specify any range for x, this is a symbolic integration, and that does not work for NIntegrate. If you replace x with a number, the result appears immediately.

If I leave off the last parameter, I get a different result, but the documentation does not cover any limits other than {x, xmin, xmax}.

There are two issues. The first is that if there is nothing to execute, then why show any results? I have input the function NIntegrate[x^a, {a, 1., 2., .1}], it should not execute, and when it is first executed, in fact I simply get back NIntegrate[x^a, {a, 1., 2., .1}]. If I re-execute an evaluation on the same function, it evaluates. Why?

My goal is to set fff=NIntegrate[x^a, {a, 1., 2., .1}] and then to use fff over variable x to do a parametric evaluation. But if, when setting fff, the function evaluates to something, what is it evaluating to and what is the function fff that I am using? I don't really have to do it this way but I did and I got an output that is ambiguous...at least to me. To my simple way of looking at things, if it should not evaluate, and so, then, it should not give back something I don't understand.

I simply want to understand what is going on and what the "false" evaluation is leaving hidden in or associated with the function fff.

OK, now that I have that off my chest, I have attached a worksheet that one of the Community members helped me develop. Do not pay attention to why I want to evaluate the particular integral, since I was going to post this to the Physics section. It is the real mutual attraction model that has never been solved....the classical point-mass integral we are all taught is a heuristic.

Consequently, I set out to prove my point. The issue became that the mutual attraction between sphere's is so close to the point-mass model that the solutions are really perturbations on the point-mass approach....which is why no issues have ever been identified. If the model is extended to cylinders and internal masses...a test mass within a distributed mass such as a sphere or disk, the results are smooth and totally unlike the point-mass results...the solutions are not simply perturbations. My goal has been to understand the detail of the spherical solutions. However, the perturbations are on the same order as the variances in the numerical results when I switch between coordinate-system descriptions of the spheres.

That being said, the worksheet models have been set up to explicitly evaluate at one point and I was attempting to generalize the approach to allow a parametric evaluation over the separation distance. It is actually easy to do and I have voluminous plots showing the result. However, when using the worksheet as a template and getting ready to post this material in the physics community, I got the behavior that I have been grumbling over and I do not understand what is going on. So far, I still don't understand the issue. To may simplistic way of looking at things, the integrals should not have evaluated at all until I defined the variable parameter...which it did in my original clunky methodology. I have no doubts about the validity of the results, but I wanted to create a more sophisticated approach and ran into these issues.

I am a bear about details, which is why I noticed that the Newtonian gravity model...or static electric interaction models as we use them...are heuristics. Now the issue is that I simply cannot understand why the numerical models evaluate at all when they are incomplete.

Regards and thanks for your forbearance.

Luther

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