MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let R be the region in the xy-plane bounded by y=x, the x-axis, and the vertical line x=4. The region R is revolved about the vertical line x=−1.
(a) Briefly describe the shape of the region R and indicate the line x=−1 on a sketch of the xy-plane.
(b) For 0≤x≤4, write expressions for the outer radius R(x) and the inner radius r(x) of a representative cross-section of the solid generated by the rotation.
(c) Set up and evaluate a definite integral that gives the volume of the solid produced. Express your final answer in exact form.