Calculating the Integral over n ball ( Please Help )

Hello ,

I am tying to write a code to calculate the value of the integral over n-ball in any dimesion n of Euclidean space using symbolic integration in (hyper)spherical coordinates

https://en.wikipedia.org/wiki/N-sphere#Spherical_coordinates

but i am confused how can i do the last part of my code and is that possiable to do it with only one loop ?

this is my code :

% multiple integrals
fun={ @(r) 1, @(r) r, @(r) exp(-r), @(r) exp(-r^2) };
R=[1 1 2 3];
results = [ 0    3.1416    4.1888    4.9348    5.2638    5.1677
            0    2.0944    3.1416    3.9478    4.3865    4.4295;
            0    3.7322    8.1260   16.9216   33.2585   61.6291;
            0    3.1412    5.5659    9.8574   17.4419   30.8130];
for k=1:4
    for n=2:6
        Int(n)=multint(n,R(k),fun{k});
    end
    Int
    if all(round(Int,4)==results(k,:))
        disp(['Case ',num2str(k),': result is correct!'])
    else
        disp(['Case ',num2str(k),': something is wrong!'])
    end
end
return

function Int=multint(n,R,fun)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Input:   n...dimension of Euclidean space En (double), n>=2         %
%          R...radius of n-ball Bn in En (double), R>0                %
%          fun...handle of function fun(r) to be integrated over Bn   %
%---------------------------------------------------------------------%
% Output:  Int...value of the integral (double)                       %
%                                                                     %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% check validity of input
Int=NaN;
if ~isreal(n) | n~=round(n) | n<2 
    warning('input n not valid, return NaN')
    return
end
if ~isreal(R) | R<=0
    warning('input R not valid, return NaN')
    return
end
if ~isa(fun,'function_handle')
    warning('input R not valid, return NaN')
    return
end
%
% I need to write my function here
%
return
end

[darova]

How fast should it be done? How urget is it?