Items tagged with triple-integral


How to find the integral of (x+y)/(x+y+z) over the part of the unit ball  centered at the origin which lies in the positive octant { x>=0 , y>=0, z>=0 } ? Numeric calculations suggest Pi/9.

How to calculate the integral of (z-z0)*z/sqrt((x-x0)^2+(y-y0)^2+(z-z0)^2)
over the unit sphere {(x,y,z):x^2+y^2+z^2<=1}
under the assumtion x0^2+y0^2+z0^2<=1 (x0^2+y0^2+z0^2>1)?
Its physical interpretation suggests the integral can be expressed through  elementary functions of the parameters.

My tries are
VectorCalculus:-int((z-z0)*z/sqrt((x-x0)^2+(y-y0)^2+(z-z0)^2),[x,y,z]=Sphere(<0,0,0>,1)) assuming x0^2+y0^2+z0^2<=1;


[r,psi,theta]=Parallelepiped(0..1,0..Pi,0..2*Pi)) assuming x0^2+y0^2+z0^2<=1;

The both are spinning on my comp. Also


is spinning.
Edt. The omitted part of the code assuming x0^2+y0^2+z0^2<=1 is added.

int(int(int(max(x^2,y^2,z^2),x=-1..1),y=-1..1),z=-1..1) with Maple?  The same question in higher dimensions.

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