Consider the region enclosed by the curves and for
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Consider the region enclosed by the curves f(x)=x2 and g(x)=2x for x≥0. This region is revolved about the horizontal line y=4 to generate a solid.
Throughout this problem, use exact values (no decimal approximations).
(a) Find the x–coordinates of the points of intersection of the two curves and state the closed interval over which the region is bounded.
(b) For a representative x in the interval you found in part (a), express the outer radius R(x) and the inner radius r(x) of a typical washer generated by the rotation. Explain verbally why each expression you give corresponds to the outer or inner radius.
(c) Write, but do not evaluate, a definite integral whose value equals the volume of the solid. Your integrand must involve the radii identified in part (b) and your limits of integration must be fully justified.
(d) The student who sets up the integral in part (c) claims that exchanging R(x) and r(x) would lead to the same numeric answer. Explain clearly why this claim is incorrect.