Consider the region in the first quadrant that is enclosed by the lines ,
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Consider the region in the first quadrant that is enclosed by the lines y=0, y=2, the curve x=(y+1)2, and the line x=y+1. This region is revolved about the y-axis to create a three-dimensional solid.
(a) For each value of y in 0≤y≤2, identify which of the two bounding curves, x=(y+1)2 or x=y+1, is farther from the y-axis. Explain your reasoning without performing any integration.
(b) Express the distance from the y-axis to the outer boundary of the region as a function R(y) and the distance to the inner boundary as a function r(y).
(c) Write, but do NOT evaluate, a single definite integral whose value equals the volume of the solid generated by the rotation described above. Clearly indicate the limits of integration and the integrand.