I am trying to express $$\iiint_Ey\,dV,$$ where $E$ is the region bounded on the sides by the cylinder $x^2+y^2=2y$; on top by $z=x^2+y^2$; and at the botton by the $xy$-plane, in the orders of $dzdrd\theta$, $drdzd\theta$, $d\theta dzdr$, and $d\theta drdz$. I was able to write the first iterated integral as $$\int_0^\pi \int_0^{2\sin\theta}\int_0^{r^2}r\sin\theta \, rdzdrd\theta.$$ I attempted the second order and was only able to get this far: $$\int\int\int_\sqrt z^{2\sin\theta}r\sin\theta \, rdrdzd\theta.$$ I am trying to figure out the bounds for the $z$ (and $\theta$) integral but I have been stumped. I also attempted the other two orders but could not get anywhere. Any guidance is appreciated,
Thanks.
THINK:
Now,
1. When we integrate with respect to $r$ first , then $\theta$ and $z$ will be constant. The intersection of $z=c$ and $\theta=c$ gives lines parallel to $xy$-plane.
To set limits of $r$ , think from where to where the lines are going within the region but the feet of lines will be on $z$-axis, because $r$ is always positive.(remember at $z$-axis , $r=0$)
2. This is similar to point $1$ , when we integrate $z$ first , then we get lines parallel to $z$-axis by intersecting $r=c$ & $\theta=c$ .
Therefore this time for the limits of $z$ , you have to decide from what place to where the lines are going within the region.. So, The limits of $r$ will be decided by those lines.($c$ stands for constant)
3. Similarily , if we take $\theta$ first, we get circular rings on $z=c$ plane in the whole region , when we intersect $z=c$ & $r=c$ .
This time $\theta $ will be decided by those rings whether they are full or what!
NOTE:
Look , I just want to give you hint about the intuition that how to imagine the region in different orders and how to deal with that. If you are still unable to think , feel free to ask , I'll post the limits in all asked orders.