area of the -sphere
Switching to polar coordinates we let and the integral becomes
The first integral is the integral over all solid angles and is exactly what we want to evaluate. Let us denote it by . With the change of variable , the second integral can be evaluated in terms of the gamma function :
We can also evaluate directly in Cartesian coordinates:
where we have used the standard Gaussian integral .
Finally, we can solve for the area
If the radius of the sphere is and not , the correct area is .
Note that this formula works only for . The first few special cases are
, hence (in this case, the area just counts the number of points in );
, hence (this is the familiar result for the circumference of the unit circle);
, hence (this is the familiar result for the area of the unit sphere);
, hence ;
, hence .
|Title||area of the -sphere|
|Date of creation||2013-03-22 13:47:06|
|Last modified on||2013-03-22 13:47:06|
|Last modified by||CWoo (3771)|