1 Find the area of the part of the surface 2z = x^2 that lies directly above the triangle in the xy-plane with the vertices at (0,0),(1,0) and (1,1).
2 Find the volume of the region in the first octant that is bounded by the hyperbolic cylinders xy = 1, xy = 9, xz = 4, yz = 1, and yz = 16. Use the transformation u = xy, v = xz, w = yz. Note that uvw = x²y²z².
The solution is comprised of detailed explanation of the application of double and triple integrals on the calculation of area and volume of general regions. It also includes necessary graph illustration and step-by-step elaborations.