example of Riemann triple integral

Determine the volume of the solid in 3 by the part of the surface

(x2+y2+z2)3= 3a3xyz

being in the first octant (a>0).

Since x2+y2+z2 is the squared distance of the point  (x,y,z)  from the origin, the solid is apparently defined by


By the definition


in the parent entry (http://planetmath.org/RiemannMultipleIntegral), the volume in the question is

V=D1𝑑v=D𝑑x𝑑y𝑑z. (1)

For calculating the integral (1) we express it by the (geographic) spherical coordinatesMathworldPlanetmath through


where the latitude angle φ of the position vector r is measured from the xy-plane (not as the colatitude ϕ from the positive z-axis); λ is the longitude.  For the change of coordinates, we need the Jacobian determinant


which is simplified to r2cosφ.  The equation of the surface attains the form

r6= 3a3r3cos2φsinφcosλsinλ,



In the solid, we have  0rr(φ,λ)  and

r= 0if only ifcos2φsinφcosλsinλ= 0.

Thus we can write


getting then


Remark.  The general for variable changing in a triple integral is

Title example of Riemann triple integral
Canonical name ExampleOfRiemannTripleIntegral
Date of creation 2013-03-22 19:10:59
Last modified on 2013-03-22 19:10:59
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Example
Classification msc 26A42
Synonym volume as triple integral
Related topic Volume2
Related topic VolumeAsIntegral
Related topic SubstitutionNotation
Related topic ChangeOfVariablesInIntegralOnMathbbRn
Related topic ExampleOfRiemannDoubleIntegral