Let be the region in the first quadrant bounded by the graph of
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let R be the region in the first quadrant bounded by the graph of x=3y−y3, the y–axis, and the line y=0. The region is revolved about the y–axis, forming a solid.
(a) Find, to exact value, the positive y–coordinate of the point where the curve x=3y−y3 intersects the y–axis other than at the origin.
(b) Write, but do not yet evaluate, a definite integral that represents the volume of the solid generated by the rotation described.
(c) Evaluate the integral obtained in part (b) to determine the exact volume of the solid.