How to calculate a surface integral on a helicoid?

Question

I have to calculate $$\iint_S \sqrt{x^2+y^2} \,dx\,dy$$ where $S$ is a 3-D helicoidal surface defined by :

\begin{align} x&=3v\cos(\theta)\\ y&=3v\sin(\theta)\\ z&=2\theta \end{align} with $$0 \le v \le 1$$ and $$0 \le \theta \le \frac{\pi}{2}$$

What I would do : replace $x$ and $y$ by there parametric representation inside the integral, change the bounds also but I am stuck with the Jacobian... How to calculate it in this case, as going from 3 to 2 variables ?

Thanks

Answer 1

You have the parametrization $r(v,\theta)=(3vcos(\theta),3vsin(\theta),2\theta)$. Now by simple calculation:

$r_v=(3cos(\theta),3sin(\theta),0)$

$r_\theta=(-3vsin(\theta),3vcos(\theta),2)$

Now you need to calculate the cross product of the vectors $r_v\times r_\theta$. Also note that $\sqrt{x^2+y^2}=3v$. So then your integral will be:

$\int_0^1\int_0^\frac{\pi}{2}3v||r_v\times r_\theta|| d\theta dv$.