A region W in the plane is bounded above by the graph of y=4−x2, bounded below by the x-axis, and bounded on the left by the y-axis. The region is revolved about the horizontal line y=d, where d>4, producing a solid whose volume depends on d.
(a) Derive, in terms of d, an explicit formula V(d) for the volume of the solid. Express your answer as a single definite integral evaluated to a closed-form expression containing d.
(b) Use your result from part (a) to determine V(6), the volume when the region is revolved about y=6. Provide the exact value.
(c) Determine the limit of V(d) as d→∞, and explain geometrically why this limit is reasonable.