I have to find the surface area of a paraboloid within a cylinder. I have to find the surface area of a paraboloid within a cylinder. The paraboloid is $x = y^2 + z^2$ and the cylinder is $y^2 + z^2 = 4$, and I know the equation but I have no idea how to set this problem up, can somebody help me with that?

Let $g(y,z)= y^2 + z^2.$ As I understand your question, you are to find the surface area for the cylindrical region $y^2+z^2\leq 4.$ Then $$ A = \int\int_{y^2+z^2\leq 4} \sqrt{1 + (g_y)^2 + (g_z)^2} \ dy \, dz. $$ Then convert to polar coordinates: $$ A = \int^{2\pi}_0 \int^2_0 \sqrt{1 + 4r^2} \ r \, dr \ d\theta $$