I am having a hard time with this question and would love to get some help. The question is:
Calculate the mass of the surface: $$ S = \left\{(x,y,z) : z = \sqrt{x^2+y^2},\ x \leq y \leq 4x -2 - x^2 \right\} $$ with density $$ \rho(x,y,z) = \frac{2z}{\sqrt{x^2+y^2}} $$
The first thing to notice is that $S$ is the graph of a function $g$ over a region $D$ in the $xy$-plane. In situations like that, surface integrals of functions can be calculated as regular two-variable area integrals: $$ \iint_S \rho\,dS = \iint_D \rho(x,y,g(x,y)) \sqrt{1+g_x^2+g_y^2}\,dA $$ The next thing to notice is that $D$ can be described as the region between the graphs of functions of $x$: $$ D = \left\{(x,y) : a \leq x \leq b,\ \phi_1(x) \leq y \leq \phi_2(x) \right\} $$ In situations like that, area integrals can be calculated as iterated integrals: $$ \iint_D \rho(x,y,g(x,y)) \sqrt{1+g_x^2+g_y^2}\,dA = \int_a^b \int_{\phi_1(x)}^{\phi_2(x)}\rho(x,y,g(x,y))\sqrt{1+g_x^2+g_y^2}\,dy\,dx $$ Now you “just” have to do these iterated integrals. But for this particular $g$ and $\rho$ things work out pretty nicely.