A region R in the xy-plane is bounded above by the parabola y=4−x2 and below by the x-axis. A solid is constructed so that every cross section perpendicular to the y-axis is a semicircle whose diameter lies in R.
Students often elect to integrate with respect to x for problems involving areas under curves. In the context of this solid, integrating with respect to y is more natural.
(a) Describe, without performing any integration, how the diameter of a representative semicircular cross section can be expressed as a function of y. Your answer should reference specific features of the region R.
(b) Explain why integrating with respect to y directly yields the volume of the solid, whereas an approach integrating with respect to x would require partitioning the base region into separate subregions or using multiple integrals. Support your explanation with clear geometric or algebraic reasoning.