1. Introduction
In this Note we focus our attention on a class of surface integrals over ndimensional real spheres which can be evaluated explicitly. We postpone the evaluation of surface integrals over generic ellipsoids because they involve transcendental elliptic integrals.
Like in [1], let Rn denote the ndimensional Euclidean space, r the position vector in Rn and r = r its norm
(1)
where the x's are its Cartesian coordinates. We shall often use ntuples of nonnegative real exponents
(2)
which, however, are not to be intended as elements of Rn.
The shorthand will be exploited in conventional expressions of the type
(3)
In this notation, an ndimensional spherical surface S(R) of radius R is defined by the condition
(4)
We are interested in the evaluation of the following integrals over S(R):
(5)
where dσ is an (n1)dimensional surface element.
2. Evaluation of the integrals
Evaluation of the integrals (5) is considerably simplified by three facts:
a) From reference [1], Equation 17, we know the value of the corresponding volume integral over the ndimensional sphere, which turns out to be
(6)
where Γ(x) denotes the gamma function [4,5].
b) Integrals (5) have the nearly selfevident scaling property
(7)
arising from the fact that E(r,p) scales with 2pth power of R and dσ scales with (n1)st power or R.
c) For spheres (unlike generic ellipsoids), the volume integration can be carried out by summing the contributions of concentric shells defined by radii r and r+dr, for r ranging from 0 to R. Hence
(8)
From (8) and (7) it follows that
(9)
from which, comparing with (6), we obtain the result
(10)
In the original paper [3] which stimulated this series of Notes, formula (10) was derived in a more direct  and also more laborious  way. It is therefore pleasing to note that the two procedures lead to the same result.
3. Admissible values of the exponents (p's)
For reasons discussed in [1], the values of p_{k} may assume any real value greater than 1/2, which are also the values compatible with Equation (10). Moreover, one should keep in mind that in equations (3) and (5), the individual Cartesian coordinates x_{k} get first squared and only afterwards elevated to p_{k}. Thus, for example, when p_{k} = 1/4, the corresponding factor in Eq.(3) is the square root of x_{k}.
4. Special cases
Setting all the p's equal to ν/2, one obtains the following formula, valid for any ν>1:
(11)
In particular, setting ν = 1 and taking into account that Γ(n) = (n1)!, we obtain the formula
(12)
When only one of the p's equals ν/2 and all the others are zero, Eq.(10) yields the identity, again valid for any ν>1:
(13)
For the important cases of ν=1 and ν=2, this gives
(14)
where
(15)
5. Surfaces of ndimensional spheres
When all the p's are zero, Eq.(10) gives simply the surface of the ndimensional sphere. Explicitly, this turns out to be
(16)
showing that the the numbers S_{n} of Eq.(22) are the surfaces of ndimensional spheres of unit radius whose values can be easily calculated by means of the recurrence
(17)
The surface of an ndimensional sphere has the dimension of the (n1)st power of length. This coincides with the popular notion of "surface area" only in the case of n=3 but its extension to any n is quite obvious. In particular, for n=2 it coincides with the common concept of "circumference" of a circle. The value of 2 for n=1, implicit in the above recurrence relations, can be considered as the cardinality of the set of the two end points of a 1D interval.
Surfaces and other surface integrals for ndimensional spheres with unit radius

Dim 
Surface_{ } 

Surface integral of:_{ } 
symbolic_{ }^{ } 
numeric_{ }^{ } 

x_{1}^{ } 
x_{1}^{2} 
x_{1}x_{2}^{ } 
x_{1}^{2}x_{2}^{2} 

1 
2 
2.000000.. 

2 
2 
 
 
2^{ } 
2π^{ } 
6.283185..^{ } 

4^{ } 
π^{ } 
2^{ } 
π/4^{ } 
3^{ } 
4π^{ } 
12.566370..^{ } 

2π^{ } 
4π/3^{ } 
8/3^{ } 
4π/15^{ } 
4^{ } 
2π^{2} 
19.739208..^{ } 

8π/3^{ } 
π^{2}/2 
π^{ } 
π^{2}/12 
5^{ } 
8π^{2}/3 
26.318945..^{ } 

π^{2} 
8π^{2}/15 
16π/15^{ } 
8π^{2}/105 
6^{ } 
π^{3} 
31.006276..^{ } 

16π^{2}/15 
π^{3}/6 
π^{2}/3 
π^{3}/48 
7^{ } 
16π^{3}/15 
33.073361..^{ } 

π^{3}/3 
16π^{3}/105 
32π^{2}/105 
16π^{3}/1185 
8^{ } 
π^{4}/3 
32.469697..^{ } 

32π^{3}/105 
π^{4}/24 
π^{3}/12 
π^{4}/240 
9^{ } 
32π^{4}/105 
29.686580..^{ } 

π^{4}/12 
32π^{4}/945 
64π^{3}/945 
32π^{4}/10395 
10^{ } 
π^{5}/12 
25.501640..^{ } 

64π^{4}/945 
π^{5}/120 
π^{4}/60 
π^{5}/1440 

Note: Numerically, the surface integral of x_{1}^{2} over a unit sphere equals its volume.
6. Asymptotic behavior
Equation (15) for the surface of an ndimensional sphere of unit radius can be rewritten in terms of the following function σ(x):
(18)
whose graph is shown on the right (bold black line), together with the analogous function ν(x) for unit sphere volumes (thin red line). The relation between the two functions is σ(x) = 2x.ν(x).
It is evident that both functions pass through a maximum and then, for large x, decay rapidly to zero. A very good approximation for the logarithms of both functions can be obtained by a straightforward application of Stirling formula [4,5] for ln(Γ(x)).
The asymptotic behavior is rather complicated. Roughly, one can affirm that, for very large x, the decay of both functions is dominated by a factor of the form O[x^{x}].
