Variable dimensional integrals

Question

I need to Integrate a function, but let the number of dimensions vary.

Integrate[Product[Exp[-( t[[i]] )^2], {i,1,4}], options]

with

t = {u,v,x,y}

and

options = {#,-Infinity,Infinity}&/@{u,v,x,y}

but the double braces on options doesn't allow this to work. I get this for options:

{{u, -Infinity, Infinity}, {v, -Infinity, Infinity}, {x, -Infinity, Infinity}, {y, -Infinity, Infinity}}

If I want a general list of variables, can these be added as variables into integrate, with the corresponding limits? I need a variable dimension, simply taken from the length of a list.

Answer 1

You can use Sequence to eliminate a set of brackets:

t = {u, v, x, y};
Integrate[
    Product[
        Exp[-( t[[i]] )^2],
        {i,1,4}
    ],
    Sequence @@ ({#,-Infinity,Infinity}&/@t)
]

π^2

However, it is simpler to use a Region specification instead:

Integrate[Exp[-p.p], p ∈ FullRegion[4]]

π^2