Computing the surface integral of a parabloid

[ainster31]

Homework Statement

Homework Equations

The Attempt at a 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?

 

[haruspex]

R is the projection of the paraboloid onto the XY plane. The surface boundary is given in the question by the lines x=0, y = x√3, z=1. Since 2z = 1 + x²+y² for the surface, that last translates into 1 = x²+y² for R. So R's boundary consists of two straight lines and an arc of a circle.
If θ is measured from the +ve x-axis, what are its values at the two straight lines?

 

[LCKurtz]

If θ is measured from the +ve x-axis, what are its values at the two straight lines?

Well Haruspex! I'm shocked, SHOCKED!

 

[haruspex]

Well Haruspex! I'm shocked, SHOCKED!

Only if you touch the -ve at the same time?

 

[ainster31]

Edit: never mind.

 

[ainster31]

Edit: never mind again.

 

[ainster31]

OK, I think I got it, but that was ridiculously complicated.