I have a parametric representation of a curve: $$\gamma : [a,b] \rightarrow (r,z) \in \mathbb R^+ \times \mathbb R$$
and I want to find a general formula of the surface area of the revolution given by: $$\phi:(u,v) \in [a,b]\times[0,2\pi] \mapsto (\gamma_r(u)\cos v, \gamma_r(u) \sin v, \gamma_z(u))$$
Since I know the formula for general surfaces is:
$$A = \int_K \Big | \Big |\frac{\partial\phi}{\partial u} (u,v) \times\frac{\partial\phi}{\partial v }(u,v) \Big | \Big | d(u,v)$$
I started by calculating the normal vector, so:
$$\frac{\partial\phi}{\partial u} (u,v)= (\gamma_r(u)'\cos v, \gamma_r(u)' \sin v, \gamma_z(u)') $$ $$\frac{\partial\phi}{\partial v} (u,v) = (-\gamma_r(u)\sin v, \gamma_r(u) \cos v, 0) $$
and by taking the cross product I got $$\frac{\partial\phi}{\partial u} (u,v) \times\frac{\partial\phi}{\partial v }(u,v) = (\gamma_z(u)'\gamma_r(u)\cos v, \gamma_r(u)\sin v, - \gamma_r(u)\gamma_r(u)')$$
This is where I got stuck however... When taking the norm and then the integral, I cannot really simplify it a lot more. I think I have to substitute for $\gamma$, but I am pretty lost right now. I do not know how to get rid of the functions $\gamma_r$ and $\gamma_z$. Any help would be greatly appreciated!
That norm is $\sqrt{\gamma_r(\gamma'_r+ \gamma'_z)}$. That is a function of u only. The "general formula" for surface area is $2\pi\int_a^b\sqrt{\gamma_r(u)(\gamma'_r+ \gamma'_z)}du$. Of course, what that is, and whether it is expressible in a simple form, depends strongly on what $\gamma_r(u)$ and $\gamma_z(u)$ are.