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IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos, CSIC-UIB, Campus UIB, E-07122 Palma de Mallorca, Spain
Received: 11 December 2009 / Accepted: 18 December 2009 / Published: 22 December 2009
The volume of the body enclosed by the n-dimensional Lamé curve defined by is computed.
A recent paper  derives asymptotic expressions for the volume of the n-dimensional body defined by for , . This is the body enclosed by a Lamé curve in n dimensions. Here I compute exactly this volume by using a straightforward modification of the calculation that gives the volume of the n-dimensional sphere, the case , see .
Writing , the volume is
By dimensional analysis . Let us now compute the integral
by using the change of variables and the volume element as
Equaling these two expressions one gets:
which is the desired formula. This validates the results in , since it coincides with the approximate calculation of that paper in the asymptotic limit although, as proven here, it turns out to be valid for any value of n.
I acknowledge financial support by the MEC (Spain) and FEDER (EU) through project FIS2007-60327.