Give the limits of integration for evaluating the integral $\iiint_Rf(r,\theta,z)\,dz\,r\,dr\,d\theta$ as an iterated integral over the region that is bounded below by the plane $z=0$, on the side by the cylinder $r=9\cos\theta$, and on top by the paraboloid $z=3r^2$.
$0\le z\le2r^2$; $0\le r\le9\cos\theta$; what about $\theta$?
What I don't understand is how to get the last one. I would think the lower limit of $ \theta$ would be $0$, and its upper limit would be $2π$, considering both components of the 3D objects have the full range of rotation for $θ$. Apparently the answer is ACTUALLY that $-\pi/2 ≤ \theta ≤ \pi /2$. I don't understand how to arrive at that answer.
The plot of your region is:
I've drawn the cylinder as a spiral in order not to cover the inside. Let's just look at the equation of your cylinder: $$r = 9 cos(\theta) $$ $$ r^2 = 9rcos(\theta)$$ Since $r^2 = x^2 + y^2$ and $x = r cos(\theta)$ we have: $$x^2 + y^2 = 9x$$ $$(x - 4.5)^2 + y^2 = 4.5^2 = r^2$$ From this we can conclude that the angle $\theta$ which lies on the $xy$ plane is bounded by $-\frac{\pi}{2}$ and $\frac{\pi}{2}$