# Mathematica®: performing a varying number of multiple integrals

1. Jul 18, 2011

### pumpf

Hello everyone.
In Mathematica® I want to numerically integrate a function of k variables (k varies) with respect to all of them. Does anyone of you know a way to do that? I tried the following simplified example.

k = 5;
int[x_] := x[[1]] + x[[2]] + x[[3]] + x[[4]] + x[[5]] ; (* My integrand. Of course, together with a and b below, the true version will be defined in terms of k. *)
a = {1, 2, 3, 4, 5}; (* lower limits of integration *)
b = {2, 3, 4, 5, 6}; (* upper limits of integration *)
For[i = k, i >= 1, i--,
y = Table[x[j], {j, 1, i}];
int[x_] = Integrate[int[y], {x, a[], b[]}] /. {x[p_] -> x[[p]]};
]

int becomes now a constant function, which is what I wanted. My problem is that my initial integrand int[x_] is more complicated than the one written in the example and is not analytically integrable, so that I have to use numerical integration NIntegrate instead of Integrate. However, I cannot NIntegrate one variable at a time. Also using the definition with := (int[x_] := Integrate[...]) trying to perform just one numerical estimation at the end is not working. Any ideas to help me with this problem?

Lucio

2. Jul 18, 2011

### Bill Simpson

It is very difficult to be certain I understand the problem you have tried to simplify.

This
http://reference.wolfram.com/mathematica/ref/NIntegrate.html
shows you can
NIntegrate[f,{x,x0,x1},{y,y0,y1},{z,z0,z1}...]

That will let you numerically integrate over all your variables in a single step.

Will that solve your problem?

3. Jul 18, 2011

### pumpf

My problem is that in the non-simplified problem the number of variables is k, with k varying in a for cycle.

4. Jul 18, 2011

### Bill Simpson

Again and again people show up here with slight variations of "My problem is too complicated to understand or explain, I absolutely positively MUST use For to do it, but it doesn't work, what do I do?" That almost never turns out well.

v={{x,x0,x1},{y,y0,y1},{z,z0,z1}};
For[i=1,i<=3,i++,
Print[NIntegrate[f,Evaluate[Sequence@@Take[v,i]]]]
]

There is more going on inside that than can be briefly explained to a new user.

I don't believe this is going to solve your problem, but good luck.

Last edited: Jul 18, 2011
5. Jul 19, 2011

### pumpf

It did solve my problem, thanks! I didn't know the commands Sequence and @@ (Apply), which turned out to do what I wanted. I programmed the k-th step the following way:

y = Table[x, {i, 1, k}]; (* my k variables *)
a = Table[i, {i, 1, k}]; (* lower limits of integration *)
b = Table[i + 1, {i, 1, k}]; (* upper limits of integration *)
v = Table[{x, a[], b[]}, {i, 1, k}];
f[x_] := E^(Sum[-x[]^2, {i, 2, k - 1}] - 0.5 (x[[1]]^2 + x[[k]]^2)); (* my integrand *)
Print[NIntegrate[f[y], Evaluate[Sequence @@ v]]];

Thanks for your time and your suggestions.

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