A solid has rectangular cross-sections perpendicular to the –axis over the interval
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A solid has rectangular cross-sections perpendicular to the x–axis over the interval 0≤x≤4. The base of each rectangle spans the region bounded by y=x and the x–axis. The height of each rectangle equals the length of its base for 0≤x≤2 and is twice the length of its base for 2<x≤4.
(a) Write a piecewise expression for the length of the base of a typical cross-section on the interval 0≤x≤4.
(b) Set up, without yet integrating, a single piecewise definite integral that represents the volume of the solid.
(c) Evaluate the integral(s) to obtain the exact volume.
(d) Explain how the change in cross-sectional proportions at x=2 influences the overall volume. Support your explanation with quantitative evidence from your calculations.