Let be the region in the first quadrant bounded above by the curve and below by the -axis
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let R be the region in the first quadrant bounded above by the curve y=2x−x2 and below by the x-axis. The region is revolved about the x-axis, producing a three-dimensional solid.
(a) Determine the x-coordinates of the points where the graph of y=2x−x2 intersects the x-axis.
(b) Write a definite integral whose value equals the volume of the solid generated by the rotation.
(c) Evaluate the integral obtained in part (b) to find the exact volume in terms of π.