1. ## Triple integral..confused about limits

Hello!

I need help with the following task:
∫∫∫ z/(1+x^2+y^2) dxdydz
D

where D is: {(x,y,z); x^2+ y^2+z^2 equal to or less than 1, z equal to or bigger than sqrt(x^2 + y^2)}

I think I need to introduce polar coordinates, but I don't understand which limits to set..can someone please help me with this? I am preparing for an exam next week and really need some guidance.

Cheers!

2. $\displaystyle x^2+ y^2+ z^2\le 1$ is inside and on the unit sphere. $\displaystyle \sqrt{x^2+ y^2}= z$ is the upper nappe of a cone so the region here is above that cone but inside the sphere. Yes, convert to polar coordinates (strictly speaking cylindrical coordinates). $\displaystyle x= r cos(\theta)$, $\displaystyle y= r sin(\theta)$, so the sphere is $\displaystyle r^2+ z^2= 1$ and the cone is $\displaystyle r= z$. The cone and the sphere intersect where $\displaystyle r^2+ z^2= r^2+ r^2= 2r^2= 1$ or $\displaystyle r= \sqrt{\frac{1}{2}}= \frac{\sqrt{2}}{2}$. The integral with respect to $\displaystyle \theta$ is from $\displaystyle 0$ to $\displaystyle 2\pi$, of course. The integral with respect to r is from 0 to $\displaystyle \frac{\sqrt{2}}{2}$ and the integral with respect to z from $\displaystyle r$ to $\displaystyle \sqrt{1- r^2}$.