Region R is enclosed by the horizontal line and the parabola for
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Region R is enclosed by the horizontal line y=4 and the parabola y=x2 for 0≤x≤2. Region R is revolved about the x–axis to create a solid.
(a) Explain, with reference to the boundaries, why a hole (empty central region) appears in every cross-section taken perpendicular to the x–axis for 0<x<2.
(b) Express the outer radius R(x) and the inner radius r(x) of a typical cross-section in terms of x.
(c) Formulate, but do not evaluate, a definite integral that gives the exact volume of the solid generated by revolving Region R about the x–axis. Show clear limits of integration and integrand.
(d) Suppose instead that Region R is revolved about the horizontal line y=−1. State how (if at all) each of the expressions you gave in part (b) would change, and briefly justify your answer.