Find the surface area of the helicoid (spiral ramp) with vector equation
\[
\bar{r}(u, v)=\langle u \cos v, u \sin v, v\rangle, 0 \leq u \leq 2,0 \leq v \leq 2 \pi .
\]

Step 1 ：Define the vector function \(\bar{r}(u, v)=\langle u \cos v, u \sin v, v\rangle\).

Step 2 ：Find the partial derivatives \(\bar{r}_u\) and \(\bar{r}_v\) of \(\bar{r}\).

Step 3 ：\(\bar{r}_u = \langle \cos v, \sin v, 0 \rangle\) and \(\bar{r}_v = \langle -u\sin v, u\cos v, 1 \rangle\).

Step 4 ：Find the cross product of \(\bar{r}_u\) and \(\bar{r}_v\).

Step 5 ：The cross product is \(\langle \sin v, -\cos v, u \rangle\).

Step 6 ：Find the magnitude of the cross product.

Step 7 ：The magnitude is \(\sqrt{u^2 + 1}\).

Step 8 ：Integrate the magnitude of the cross product over the given region.

Step 9 ：The surface area of the helicoid is \(2\pi\left(\frac{{\sinh^{-1}(2)}}{2} + \sqrt{5}\right)\).

Step 10 ：\(\boxed{2\pi\left(\frac{{\sinh^{-1}(2)}}{2} + \sqrt{5}\right)}\) is the final answer.