Items tagged with triple-integral

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I am trying to evaluate the following triple integral but it takes much time so i kill the job.


 

restart; R := 5; KK := proc (theta) options operator, arrow; evalf(int(int(int(1/(R*sin(theta)^2+(R*cos(theta)+Z)^2+(2*R*k.sin(theta))*cos(p))^2, p = 0 .. 2*Pi), Z = 0 .. 60), k = 1 .. 10, numeric)) end proc; evalf(KK((1/6)*Pi))

Warning,  computation interrupted

 

``


 

Download int_maple_prime2.mw

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;

and

VectorCalculus:-int(eval((z-z0)*z/sqrt((x-x0)^2+(y-y0)^2+(z-z0)^2),
[x=r*sin(psi)*cos(theta),y=r*cos(psi)*sin(theta),z=r*cos(psi)])*r^2*sin(psi),
[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

VectorCalculus:-int((z-1/4)*z/sqrt((x-1/2)^2+(y-1/3)^2+(z-1/4)^2),[x,y,z]=Sphere(<0,0,0>,1),numeric);

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

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