Final answer:
To find the volume of the given surface z = 16 − x^2 - y^2, we need to set up a double integral over the region in the xy-plane where the surface has a positive z-value. The volume of the surface is 256π/3 cubic units.
Step-by-step explanation:
To find the volume of the given surface, z = 16 − x^2 - y^2, we need to set up a double integral over the region in the xy-plane where the surface has a positive z-value. Since z = 0 is the xy-plane, we can set up the integral as follows:
Volume = ∬ D z dA
where D is the region in the xy-plane bounded by the curves x^2 + y^2 = 16. We can express this region in polar coordinates as:
D = 0 ≤ r ≤ 4, 0 ≤ θ ≤ 2π
Converting to polar coordinates, the double integral becomes:
Volume = ∫02π ∫04 (16 - r^2) r dr dθ
Simplifying and evaluating the integral, we find that the volume of the surface is 256π/3 cubic units.