MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let g be a continuous, non-negative function on the interval [1,5] with g(1)=0 and g(5)=0. A region R is bounded above by the graph of y=g(x), below by the x-axis, and on the sides by x=1 and x=5. The region is revolved about the x-axis, generating a solid S.
(a) Describe, in words, the cross-sections of the solid S taken perpendicular to the x-axis. Include the geometric shape and how its dimensions are determined by g. Explain why these cross-sections completely characterize the volume of S.
(b) Write a definite integral whose value equals the volume of S. Clearly indicate the limits of integration and the integrand, and justify each element of your expression without evaluating the integral.
(c) Suppose g is twice differentiable and attains its maximum value at x=3. Explain how the second derivative test applied to g at x=3 provides information about the maximal cross-sectional area of S.