# Mathematica®: performing a varying number of multiple integrals

• Mathematica
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

Related MATLAB, Maple, Mathematica, LaTeX News on Phys.org
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.

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

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[[email protected]@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:
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]]];