Surface Integral: Evaluating zdS in Hyperboloid of Two Sheets

[bosox284]

Homework Statement
Evaluate the surface integral, the double integral of zdS if the region is the patch of surface defined by x^2 + y^2 - z^2 = -1 in the first octant with z less than or equal to 4.


The attempt at a solution
I really don't know where to begin. I believe the equation is of a hyperboloid of two sheets. So I want to say I have to parametrize the equation with x = bxcos(u)sinh(v), y = bysin(u)sinh(v), z = bzcosh(v), and then take the partial derivatives of u and v, and take the cross product of the two. Could someone please point me in the right direction? Or instruct me where to go from there?

 

[gabbagabbahey]

The attempt at a solution
I really don't know where to begin. I believe the equation is of a hyperboloid of two sheets. So I want to say I have to parametrize the equation with x = bxcos(u)sinh(v), y = bysin(u)sinh(v), z = bzcosh(v), and then take the partial derivatives of u and v, and take the cross product of the two. Could someone please point me in the right direction? Or instruct me where to go from there?

That sounds like a good plan, although I see no need for the $b$'s...why not show us how far you get with that method?

 

[bosox284]

Well the b's would be for if it were like (x^2)/4, where b would be 2. So in this case, the value of b is one and I could have gone without them. The only problem I see myself running into from here, would probably be finding out the values of u and v for when I set up the integral.

 

[gabbagabbahey]

That shouldn't be too difficult...$x$,$y$, and $z$ all have to be positive in the first octant and $z\leq 4$...find the range on $v$ first using your limits on $z$ and then see what range of $u$-values $x$ and $y$ are positive for.