Final answer:
The question involves calculus to calculate the average height of a hemispherical surface over a disk in the xy-plane by setting up a double integral and normalizing by the disk's area.
Step-by-step explanation:
The question is about calculating the average height of the hemispherical surface given by the equation z = d√(a² - x² - y²) above the disk x² + y² ≤ a² in the xy-plane. The subject falls under the category of calculus, specifically in the field of solid geometry with an application of integrals to find average values over a region. To find the average height of the hemisphere, one would integrate the function representing the height over the area of the disk and then divide by the area of the disk.
The formula for the hemisphere's average height z over the disk in the xy-plane will involve setting up a double integral over the circular region with radius a, integrating the height function, and normalizing it by the area of the disk, which is πa².