MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A semicircular region T is defined by (x−2)2+y2≤4 with x≥0. The region T is revolved about the vertical line x=−1.
(a) Express the volume of the resulting solid as a single definite integral with respect to y. Give explicit formulas for the outer and inner radii in terms of y, and state the limits of integration.
(b) Evaluate the integral from part (a) to find the exact volume of the solid.
(c) Write an alternative definite integral with respect to x that also represents the volume. Evaluate this integral and confirm that it yields the same result obtained in part (b).