A region in the first quadrant is bounded on the left by the –axis
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A region R in the first quadrant is bounded on the left by the y–axis, above by the horizontal line y=8, and below by the curve y=x3. The region is revolved about the horizontal line y=9 to generate a solid of revolution.
(a) Describe geometrically why every horizontal cross-section of the solid is a circular ring (an annulus). Your explanation should reference the given curves and the axis of rotation.
(b) Write, but do not evaluate, an integral with respect to x that gives the exact volume of the solid.
(c) Express x as a function of y, and then write, but do not evaluate, an integral with respect to y that also gives the exact volume of the same solid.
(d) Compare the two integrals from parts (b) and (c). Explain which one involves a simpler integrand or limits of integration, and justify why both integrals must yield identical numerical results.