Final answer:
The surface area of the part of the cylinder x^2 + z^2 = 64 above the square is found by multiplying the circumference of the cylinder (2πr) by the height which corresponds to the side length of the square (4). The surface area is 64π square units.
Step-by-step explanation:
To find the area of the surface of the part of the cylinder x^2 + z^2 = 64 that lies above the square with vertices (0, 0), (4, 0), (0, 4), and (4, 4), we can combine elements of circular and rectangular geometries. Since x^2 + z^2 = 64 represents a cylinder with radius 8 (because 64 is r^2), the cylinder's circumference is 2πr. In this case, we consider only the part of the cylinder that is above the square area of sides 4 units long.
The external surface area of this portion of the cylinder can be calculated by taking the perimeter of the circle times the height (which is equivalent to the side length of the square because the surface lies directly above the square). The perimeter of the circle is the circumference of the base circle of the cylinder, which in this case is 2π(8).
The surface area is then the circumference times the height (or side length of the square), yielding 2π(8) * 4. Therefore, the surface area is 2π(8) * 4 = 64π square units.