Final answer:
The surface area of the part of the sphere x²+y²+z²=4z that lies within the paraboloid z=x²+y² is found by integrating over the region of their intersection, a task which typically employs techniques from multivariate calculus.
Step-by-step explanation:
The student has asked to find the surface area of the part of the sphere x²+y²+z²=4z that lies inside the paraboloid z=x²+y². To compute the surface area, we must first complete the square for the sphere's equation to get it into standard form. By doing so, we find that the sphere's center is at (0, 0, 2) and the radius is 2. Since the paraboloid z=x²+y² is a surface that opens upward, the intersection with the sphere will create a cap on the sphere. The surface area of the spherical cap can be computed using integral calculus, taking into account the limits set by the paraboloid.
To integrate and find the exact area, you'd set up an integral with appropriate bounds that represent the region of overlap between the two surfaces. These bounds are determined by the points where the two surfaces intersect. Please note, the method for setting up and evaluating the integral can be quite advanced, often requiring multivariate calculus techniques.