MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Consider the region S enclosed by the curve x=y3−3y, the vertical line x=0, and the horizontal lines y=−1 and y=2. The region S is revolved about the horizontal line y=−2 to create a solid of revolution.
(a) Before setting up any integrals, justify whether integrating with respect to x or integrating with respect to y leads to a simpler expression for the volume of the solid. Base your reasoning on the behavior of the radii and the need (or lack thereof) to invert the given curve.
(b) Using the variable of integration you selected in part (a), write a definite integral that represents the volume and evaluate it exactly.
(c) Without computing, write the corresponding definite integral using the other variable of integration. Explain clearly what algebraic obstacle or additional complication would arise if one attempted to evaluate this alternative integral directly.