calculus and analysis - Multiple integrals where the number of integrals is aribtrary

I know that I can do, say, a triple integral such as $$\int_{a_x}^{b_x}\int_{a_y}^{b_y}\int_{a_z}^{b_z}f(x,y,z)\,dz\,dy\,dx$$ with the input Integrate[f[x,y,z],{x,ax,bx},{y,ay,by},{z,az,bz}].

---

Now, however, I would like to do a multiple integral of the form $$\int_{a_1}^{b_1}\cdots\int_{a_n}^{b_n}f(x_1,\ldots,x_n)\,dx_n\,\ldots\,dx_1$$ for some $n$ that I specify. Is there a concise functional way to input this into Mathematica, or is the only way to manually type the full Integrate[f[x1,...,xn],{x1,a1,b1},...,{xn,an,bn}]. command?

Answer

Here is another way:

multi[f_, x_, a_, b_, n_] := 
 Inactive@Integrate @@ {f @@ Array[x[#] &, {n}]}~Join~Array[{x[#], a[#], b[#]} &, {n}]

I used Inactive because I don't know if the purpose is purely typesetting, or if you want it to try to evaluate (if so remove it, or use Activate later).

multi[g, y, ymin, ymax, 10]
(* Inactive[Integrate][
 g[y[1], y[2], y[3], y[4], y[5], y[6], y[7], y[8], y[9], 

  y[10]], {y[1], ymin[1], ymax[1]}, {y[2], ymin[2], ymax[2]}, {y[3], 
  ymin[3], ymax[3]}, {y[4], ymin[4], ymax[4]}, {y[5], ymin[5], 
  ymax[5]}, {y[6], ymin[6], ymax[6]}, {y[7], ymin[7], ymax[7]}, {y[8],
   ymin[8], ymax[8]}, {y[9], ymin[9], ymax[9]}, {y[10], ymin[10], 
  ymax[10]}] *)

perhaps a screenshot here is better:

If its just for typesetting then you can swap subscripts into it as follows: