Let R be the region in the first quadrant bounded above by the curve
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let R be the region in the first quadrant bounded above by the curve y=x, bounded below by the x–axis, and bounded on the right by the vertical line x=4. The region R is revolved about the x–axis to generate a solid S.
(1) Part 1: (a) Describe in words the geometric shape of a typical cross-section of S taken by slicing perpendicular to the x–axis at a value x with 0≤x≤4.
(2) Part 2: (b) Determine whether these cross-sections should be treated as circular disks or as washers. Justify your choice by discussing the existence (or absence) of an inner radius for all x in the interval.
(3) Part 3: (c) Write an expression (do not evaluate) for the radius or radii of the cross-section as a function of x, clearly indicating any inner or outer radius you identified in part (b).