How do I set up an d dimensional integral for general d?

*To*: mathgroup at smc.vnet.net*Subject*: [mg127262] How do I set up an d dimensional integral for general d?*From*: Jonathan Frazer <J.Frazer at sussex.ac.uk>*Date*: Wed, 11 Jul 2012 18:24:12 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net*References*: <20120711061823.1AE5D684B@smc.vnet.net>

Hi all, I can't work out how to do summations, integrals etc. for d parameters without having to rewrite the expression on a case by case basis. I have had this trouble a few times but in this instance I would like to calculate the surface area of a d-sphere, where I want to specify the number of dimensions d at the beginning but not have to change anything else. The way I was planning to do it was as follows: d = 2; =E8=E1 = Table[Symbol["=E8" <> ToString[i]], {i, d - 1}]; S=E8=E1 = Sin[=E8=E1]; C=E8=E1 = Cos[=E8=E1]; s=E8 = r Prepend[Table[Product[S=E8=E1[[=E1]], {=E1, i}], {i, d - 1}], 1]*Append[C=E8=E1, 1]; =EB = Outer[D[#1, #2] &, s=E8, =E8=E1]; g = Table[Sum[=EB[[=E3, =E1]] =EB[[=E3, =E2]], {=E3, d}], {=E1, d - 1}, {=E2, d - 1}]; vol = Sqrt[Det[g]] (*Now all I should have to do is integrate vol but I don't know how to do it \ for a general d.For instance,for d=3 write*) Integrate[vol, {=E81, 0, =F0}, {=E82, 0, 2 =F0}] (*but if d=2 I would need to write*) Integrate[vol, {=E81, 0, 2 =F0}] How do I get around this? Many thanks in advance, Jonny

**References**:**Re: Epilog and ListPlot3D***From:*Bill Rowe <readnews@sbcglobal.net>