@MichaelE2 I'm confused by the punctuation and decimals in your comment.
To avoid ambiguity. compare `With[ {a= 1}, Precision@Integrate[Sin[x], {x, a,a}] ]` and `With[ {a= 1}, Precision@Integrate[Sin[x], {x, a,N@a}] ]`
The first should be always Infinity, and the later MachinePrecision
But this should also be `Infinity` With[ {a= N@1}, Precision@Integrate[Sin[x], {x, a,a}] ]
I don't see how is justified to have a Real function integrated in a `Real` domain jumping into the `Complex` domain in `Integrate[Sin[x], {x, 1.000001`6, 1.000003`6}] (* 0.*10^-6 + 0.*10^-12 I *)`
Also, this shouldn't be `Infinity`: `Precision@Integrate[Sin[x], {x, 1.0000001`6, 1.000003`6}]`
@rhermans The "get 0." followed by a comma is a machine zero. The next "not 0" is an exact (Integer) zero followed by a period. Nothing else seems ambiguous to me, but if I left something out, ask.
@rhermans I disagree that the 3rd With[{a = N@1}...] should have infinite precision and cited the documentation to support it. I understood your mathematical argument why 0 might be considered the desired result, but that is not how Mathematica is supposed to perform. (I disagree with one of your hypotheses in that argument, that all instances of floating-point 1. represent exactly the same number and should be so treated.)
@rhermans Re the complex result: Mathematically the real numbers are a subset of the complexes. So the answer has the correct value even if we agree that the form is incorrect. However, Integrate[] computes complex integrals by default (that is, integrals over the complex domain). There are lots of questions, even complaints, about that in Q&As on the site. The following constrains x to be real:
Oops, I meant to include the following analogous problem with Solve. By your argument, the answer should be Infinity; by mine and the one in the docs, it should be MachinePrecision:
With[{a = N@1}, Precision@Solve[x + a == a, x]]
(* MachinePrecision *)
