Region P lies in the first quadrant and is bounded on the right by the parabola
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Region P lies in the first quadrant and is bounded on the right by the parabola y=x2, on the left by the parabola y=6−x2, and above and below by the x-axis whenever applicable. The region is revolved about the y-axis.
(a) Express each bounding curve as x in terms of y.
(b) Determine the y-coordinate(s) at which the two parabolas intersect and explain why these value(s) separate sub-regions within P.
(c) Write a complete set of definite integral(s) with respect to y whose sum equals the volume of the solid obtained by the revolution described.
(d) Evaluate the integral expression(s) from part (c) to find the exact volume.