A solid has its base region in the -plane bounded above by and below by the -axis for
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A solid has its base region R in the xy-plane bounded above by y=4−x2 and below by the x-axis for −2≤x≤2. Cross sections taken perpendicular to the x-axis alternate in the following way while moving from left to right:
• For −2≤x<−1 and for 0≤x<1, each cross section is a square whose side length equals the distance between the curves that bound R at that x-value.
• For −1≤x<0 and for 1≤x≤2, each cross section is a rectangle whose height is twice its base, where the base equals the distance between the bounding curves at that x-value.
The solid is created by joining the resulting cross sections flush at their boundaries.
(a) Explain how the change in cross-sectional shape at x=−1,0, and 1 affects the way the volume of the solid must be represented algebraically.
(b) Describe, without performing any integration, the separate integrals that would have to be evaluated to compute the total volume.
(c) Discuss whether the volume density (cross-sectional area as a function of x) is continuous at each transition point x=−1,0,1. Justify your conclusion in each case.