Here is a question i encounter in my homework, since it is my first time doing this kind of question, I just want to know if I have done this correctly.
Integrate the function $f(x,y,z)=x+y+z$ over the surface that is described as follows: $$x=2u-v, y=v+2u, z=v-u$$where $u\in[0,20], v\in[0,21] $
I let $S$ to be the surface so the integral should be $\int\int_Sx+y+z dS$. I first try to compute $dS$ in terms of $dudv$ with the fact that $dS=||\frac{\partial r}{\partial u}$x$\frac{\partial r}{\partial v}||dudv$ and i get that $dS=\sqrt34dudv$. Therefore, by plugging in $x,y,z$ in terms of $u,v$ and I get the integral $\int^{21}_0\int^{20}_0(\sqrt34)(3u+v)dudv$. I wold like to know if so far I have done anything wrong so that i can continue to evaluate the integral. Thank you.