Find the surface area of the helicoid (i.e., spiral ramp) given by the vector parametric equation

$$\boldsymbol{\vec{r}} \left( u, v \right) = \langle u \cos v, u \sin v, v \rangle, \ \ \ 0 \leq u \leq 1, \ \ \ 0 \leq v \leq 3 \pi.$$
(a) Find $\| \boldsymbol{\vec{r}}_u \times \boldsymbol{\vec{r}}_v \|$ and simplify your answer as much as possible.

$\| \boldsymbol{\vec{r}}_u \times \boldsymbol{\vec{r}}_v \| =$

(b) Set up and evaluate a double integral for the surface area of this helicoid. (When you answer part (a) correctly, a hint will appear here with an integral formula.)

Surface area =