Suppose you have to evaluate the surface integral $$\int\int_S (x^2+y^2+4)\space dS$$ where $S$ is the surface parameterized by $\textbf{r} = <2uv, u^2-v^2, u^2+v^2>$ with $u^2+v^2 \le 16.$
I know the equation to solve the surface integral is $$\int\int_S f \space dS = \int\int_S f(\textbf{r}(u,v))\space |\textbf{r}_u \times \textbf{r}_v |\space du \space dv$$ and I will have no trouble taking the cross product or evaluating the integral. However, how would one go about finding the limits for $S$? I don't even remotely know what $\textbf{r} = <2uv, u^2-v^2, u^2+v^2>$ might be. Is there some way maybe to transform $\textbf{r}$ to an explicit $z = f(x,y)$?
Thanks.
You do not need know how the surface is. Notice that the domain of the parametrizations is a disc of radius 4.
Hint: Since the domain is a disc, try integrate the right hand double integral by using polar coordinates.
I hope I have helped you.