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Triple integration takes long time?

Posted 1 year ago

When I compute the following triple integral Mathematica sticks running. I waited for over an hour, but it was still running. I'm a beginner, so I guess I'm making some errors in the code. That's my code: https://www.wolframcloud.com/obj/744b5d5c-e605-4e87-8fc6-24a0cbe3044f

Thanks. Giulio

POSTED BY: Giulio Giamello
Posted 1 year ago

Your triple integration is

Integrate[xy*sqrt(x^2+y^2+z^2),Element[{x,y,z},
  ImplicitRegion[z>=sqrt(x^2+y^2)&&x^2+y^2+z^2<=1,{x,y,z}]]] 

While this might not get you an answer, you have some syntax errors that need fixing. You need to use

Sqrt[...]

rather than

sqrt(...)

And xy needs a space between the x and y.

If we Reduce the restrictions, we can end up with the limits of integration:

Reduce[{z > Sqrt[x^2 + y^2], x^2 + y^2 + z^2 < 1}]
(* (0 < z <= 1/Sqrt[2] && -Sqrt[z^2] < y < Sqrt[z^2] &&
        -Sqrt[-y^2 + z^2] < x < Sqrt[-y^2 + z^2]) || 
    (1/Sqrt[2] <  z < 1 && -Sqrt[1 - z^2] < y < Sqrt[1 - z^2] &&
        -Sqrt[1 - y^2 - z^2] < x < Sqrt[1 - y^2 - z^2]) *)

And the integral is expressed as follows:

Integrate[x y*Sqrt[x^2 + y^2 + z^2], {z, 0, 1/Sqrt[2]}, {y, -z, z}, 
    {x, -Sqrt[z^2 - y^2], Sqrt[z^2 - y^2]}] +
Integrate[x y*Sqrt[x^2 + y^2 + z^2], {z, 1/Sqrt[2], 1}, {y, -Sqrt[1 - z^2], Sqrt[1 - z^2]}, 
    {x, -Sqrt[1 - y^2 - z^2], Sqrt[1 - y^2 - z^2]}]
(* 0 + 0 = 0 *)
POSTED BY: Jim Baldwin
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