Given the parametric surface $\Phi(\rho,\theta)=(\rho\cos\theta,\rho\sin\theta,\theta)$, $\rho\in]0,1]$, $\theta\in[0,4\pi]$ find its area.
This is a kind of exercise i might find in my multivariable calculus exam.
I'm struggling to find a way to solve it because there's no resolution method mentioned in the study material my prof gave me for this kind of exercise.
I guess i need to do some sort of integration to solve this, but i don't know which one to look up for. I think an example could help me understand.
Call ${\bf r}(\rho,\theta) = \rho\cos\theta \hat{x}+ \rho\sin\theta\hat{y} + \theta\hat{z}$, and define the vectors
\begin{eqnarray} {\bf r}_\rho &=& \frac{\partial {\bf r}}{\partial \rho} = \cos\theta\hat{x} + \sin\theta\hat{y} \\ {\bf r}_\theta &=& \frac{\partial {\bf r}}{\partial \theta} = -\rho\sin\theta\hat{x} + \rho\cos\theta\hat{y} + \hat{z} \end{eqnarray}
All you need to do now is to calculate $|{\bf r}_\rho\times {\bf r}_\theta|$, and integrate
$$ \int_0^1{\rm d}\rho\int_0^{4\pi}{\rm d}\theta ~|{\bf r}_\rho\times {\bf r}_\theta| $$