# Integral upper limit a function

GROUPS:
 I am new to Mathematica and am having trouble performing a general multiple integral.My issue is that the upper limit of the first of a triple integral is a function of the next integration variable.Mathmatical 9 gives several error messages. First, it says that it can't tell if the limit function is real or not. Second, it gives partial answer as a conditional with half a page of real and imaginary constraints.I cannot deternine how to tell Mathematica that all values and functions are real.I have checked the syntax multiple times, which is no guarantee of not missing something.Simple numerical limits show that the integral is not transcendental and gives real values.I have looked at a half dozen sources to determine how to trouble shoot the integral and how to insert assumptions, but nothing seems to fix the issue.Any pointers would be welcome.
4 years ago
6 Replies
 Sean Clarke 1 Vote Hi. Can you give a simple example of the kind of integral you mention?  A multiple integral where "the upper limit of the first of a triple integral is a function of the next integration variable. If I try to express this as an iterated integral, do you mean something like this:Integrate[Integrate[f[x, y], {x, 0, 1}], {y, 0, g[x]}]orIntegrate[f[x, y], {y, 0, g[x]}, {x, 0, 1}]You can use the Assumptions option to inform most functions that some constants are supposed to be real. Please see the tutorial on using Assumptions.If possible, please try giving the smallest example possible of the issue you are seeing.
4 years ago
 I looked at the Assumptions and the Reals useage, but cannot get it to work when I am doing certain integrals. When I re-did the integral again that has been troubling, I found that I had misstated my issue to you before, though both Assumptions and Reals is an issue for me in simplifying the integration times and repetitions.When I perform Integragte[r1 r2/Sqrt[r1^2+r2^2+d+r1 r2 W],{r1,0,Sqrt[a-z1^2]},{r2,0,Sqrt[a-z2^2]}], the first integral is calulated but the second is not. It is left undone, which usualy means it cannot be done, except that it can be done by hand if the limit d>r1+r2 is involked. However, I have to keep the Sqrt limits regardless of my approach.That is what I cannot get to work.This integral is further integrated over the z1 and z2 variables, which have simple limits, and there is a final double integral over W(t1,t2). I am trying to find a way of doing the complete integration, but it may not be possible without expanding the integrand as a power series and performing a less complex integral term by term. I have had some success with NIntegrate when I can use numerical limits, but not always, since I need to invoke some Assumptions about the limits being real. The equation is a parametric model, and I want to be able to vary d to produce a plot of the integral value vs d.
4 years ago
 Shenghui Yang 1 Vote Well, Mathematica does a very general check of the expression to find any part that it can simplify. So if your expression contains too many radicals and Log function, it would be fairly difficult to handle since you might not be aware of the really complicated conditions which requires too much computation. A good idea is always to simplify this type of problem at first step. At first glance the up limit of the integral can be simplified. They do not have dependance. I will just go ahead the do the integral separatedly:Integrate[x y/Sqrt[x^2 + y^2 + d + x*y*w], x]Then I will have a result. Use Rule and Replace functions to find the value of the first step integral (I just copy and paste the results): Then you can copy the above two expressions into the new integrate function:Integrate[ y (Sqrt[a^2 + d + a w y + y^2] -      1/2 w y Log[2 a + w y + 2 Sqrt[a^2 + d + a w y + y^2]]) -   y (Sqrt[d + y^2] - 1/2 w y Log[w y + 2 Sqrt[d + y^2]]), y]The result isFinally, if you really want to have the up limits to be your version, you just need to use the rule function again: /.{a->XXX,b->YYY} The problem you can see here is that once Mathematica does the symbolic computation, it basically keeps everything there and perhaps more than that you are aware of. Moreover the dependances of the radicals and wrapped Log and fractional power really makes a direct way cheesy.