Integrate f(x,y,z)=z over the domain 0

Question

Integrate f(x,y,z)=z over the domain 0<=z<=x^2+y^2<=9

anonymous · Answer

Domain is $$0 \le z \le x^2 + y^2 \le 9 $$

anonymous · Answer

I am also convert everything to cylindrical coordinates before I integrate. I am unsure as to what the limits of integration should be. I am thinking that z goes from 0 to r^2 and r goes form 0 to 3

kainui · Answer

I suggest drawing out the region of integration, that'll help a lot I think in making the other limits of integration perfectly clear.

kainui · Answer

Another helpful tip, cut up that weird thing of limits of $0 \le z \le x^2+y^2 \le 9$ into separate pieces:

$$0 \le z$$$$z \le x^2+y^2$$$$x^2+y^2 \le 9$$

anonymous · Answer

Is the region a paraboloid from z=0 to z=9?

anonymous · Answer

The region is indeed a paraboloid. Suppose we set $y=0$, then $z=x^2$ (a parabola in the $x$-$z$ plane); suppose $x=0$, then $z=y^2$ (the same but in the $y$-$z$ plane). For any value of $0\le z\le9$, you get the equation of a circle with radius $\sqrt z$. See the sketch (not drawn to scale):|dw:1449273351913:dw|