Problem:
Solution:
I am having difficulty understanding how the author determined the limits of integration of $R$. The author used $\theta=\pi/3\quad to\quad \theta=\pi/2$ and $r=1\quad to\quad r=1$. More accurately, I'm not even sure how the author graphed $R$ in the diagram. Where did he get the shape of $R$ from?
@user1251385:
Firstly, $r_{max} = 2z_{max} - 1 = 2 - 1 = 1$. Hence the upper bound is $r=1$. Then the azimuthal angle is bounded below by $\tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$ and the upper bound is the $y$-axis that is $\phi = \frac{\pi}{2}$.
Region $R$ is created by bounding the $z=1$ plane by the line $y=\sqrt{3}x$ and by letting $x^2+y^2=1$; the author is showing one edge of the surface by projecting the line $x^2+y^2=1$ from the $xy$-plane onto the paraboloid.