Final answer:
To find the area of the surface s, subtract the area of the circle from the total surface area of the sphere. The area of s is 15π square units. The center of s is (0, 0, 2).
Step-by-step explanation:
To find the area of the surface s, we first need to determine the equation for the circle that represents the intersection between the sphere and the cylinder. From the equation of the cylinder x²+y²=2x, we can complete the square to get (x-1)²+y²=1. This represents a circle with center (1,0) and radius 1.
The area of the circle is given by A = πr², where r is the radius. So in this case, A = π(1)² = π square units. Since s is the part of the sphere inside this circle and above the xy-plane, we can calculate the area of s by subtracting the area of the circle from the total surface area of the sphere.
The total surface area of the sphere is given by A = 4πr², where r is the radius. In this case, the radius is 2, so the total surface area of the sphere is A = 4π(2)² = 16π square units. Therefore, the area of s is 16π - π = 15π square units.
To find the center of s, we note that it lies on the yz-plane, which is the plane where x=0. Therefore, the center of s is (0, 0, 2).