Tuesday, 21 August 2018

integration - Any solution for intintintfracx2+2y2x2+4y2+z2,dv



I tried to solve this triple integral but couldn't integrate the result.
x2+2y2x2+4y2+z2dv and the surface to integrate in is x2+y2+z21
Is there any way to transform the integral into polar coordinates?


Answer



Notice that:
x2+2y2x2+4y2+z2dv=x2+4y2+z22y2z2x2+4y2+z2dv=1z2+2y2x2+4y2+z2dv
But by symmetry xz, we have:
x2+2y2x2+4y2+z2dv=z2+2y2x2+4y2+z2dv
So:
x2+2y2x2+4y2+z2dv=121dv=1243π=23π


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