Let be the region in the –plane that is bounded above by the parabola and below by…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let U be the region in the xy–plane that is bounded above by the parabola y=4−x2 and below by the x–axis. The region is revolved about the horizontal line y=k, where k is a fixed real number satisfying k≤0. A solid of revolution is thereby generated.
(a) Derive an explicit expression V(k) for the exact volume of the solid as a function of k. Express your answer in simplest form, valid for all k≤0.
(b) Compute V(−1).
(c) Determine the value(s) of k≤0 for which the volume of the solid is exactly twice the value found in part (b). Show all steps leading to your conclusion and discuss whether more than one such value can occur.