Question

Some important n-dimensional volumes A number that comes up frequently is the n-dimension al volume of the unit ball Bn in R" That's the integral Voly(Bu) = 11 • Use Fubini's theorem to express Voln (Bn) as -1 B(In) where B(2n) is the set of 21, 22, ..., In-1) ER-1 for which (21,22, ..., 2n-1,xn) e Bn Express B(xn) as a scaled Bn-1 CRP-1, [You might want to express (1= | 1 Bn (-1,1]" as is the actual definition, as in the plane, where 1*(P) = {0 P& Bn 1* (P) = is the characteristic function of Bn (or in dicator function). Then what is (1*)?,?]

Answer 1

Bn Answer . A number the comes up frequently n. Linensional Value Bn in k voln (B) = fe i (ux avu) and noting YER, defens f" (A) = F" (m, 2 - 2 ) FL*, *. -*-4) The orom says if fE L (exe) . Fu El (2). fy. El (M) a lady in z anty Sid (4x 2) = S(SFG. y) av (4)] de (1) = Sess Cosy) du 6-)] du ca] St=sed (w xpx-** ) n ling ::$55.6...) la tony du lund Pn. Hun

= S(td (19.] Se (92 * Therore x iphy Y =R and he