Hello everybody,

I am looking to solve a problem involving a triple (volume) integral. I intend to evaluate: $\displaystyle \iiint_{V}z(x^2+y^2+z^2)dxdydz$

where V is the upper hemisphere $\displaystyle 0 \leq r \leq a$ and $\displaystyle z \geq 0$, with $\displaystyle r = \sqrt{x^2+y^2+z^2}$.

I wish to solve using both cartesian coordinates and then spherical polar coordinates, show that the result is equal. What I wish to ask of you is, how can I determine the limits of the integration for the cartesian coordinates. I don't know how I can differentiate between a hemisphere and cylinder in this respect, since in both cases the xy-plane is a circle and then the limits for z is just 0 to a.

But, to be honest, either way I don't think I can accurately determine the limits. I am having trouble picturing it. I know you have to take one variable at a time, and the limits will be a function of the 'remaining' variables, working down to no variables in the solution.

I have tried to rearrange $\displaystyle 0 \leq r \leq a$ to get limits with one variable, say x, but then I end up with negative root on the left, which can't be right. So do I leave the lower limit as 0? Either way, I can't justify this. There may be just one simple idea I am missing.

Thank you