A region in the xy–plane is bounded above by the line and below by the line for
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A region in the xy–plane is bounded above by the line y=3−x and below by the line y=2x for 0≤x≤2. The region is revolved about the x–axis, forming a solid.
(a) Describe the shape of a typical cross-section perpendicular to the x–axis after the rotation. Identify and clearly state expressions for both the outer radius R(x) and the inner radius r(x) of that cross-section as functions of x on the interval 0≤x≤2. Give a brief justification for each choice.
(b) Explain why subtracting the square of the inner radius from the square of the outer radius at each x and integrating over 0≤x≤2 produces the exact volume of the solid. Your explanation should refer to the geometry of the cross-sections and the limiting process of integration.