Let be the region in the –plane bounded by the curves and
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let R be the region in the xy–plane bounded by the curves y=x2 and y=2x−x2. The region is revolved about the horizontal line y=−1, generating a solid of revolution.
(a) Determine the x–coordinates of all points where the two bounding curves intersect.
(b) Write, but do not evaluate, a definite integral that represents the volume of the solid obtained by revolving R about y=−1. Clearly indicate the outer and inner radii in your integrand.
(c) Evaluate the integral from part (b) to obtain the exact volume of the solid, simplifying completely.