Let Solid R be constructed with square cross sections taken perpendicular to the -axis
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let Solid R be constructed with square cross sections taken perpendicular to the x-axis. The base of Solid R is the region in the xy-plane enclosed by the graphs of y=∣x∣ and y=3−∣x∣.
(a) Sketch the two bounding graphs on the same set of axes and label clearly the points where they intersect the x-axis.
(b) Explain why the side length of a typical square cross section can be expressed as s(x)=3−2∣x∣ for −3≤x≤3. Justify the absolute-value expression in your reasoning.
(c) The function s(x) is not differentiable at x=0. Discuss whether this lack of differentiability affects the existence of the definite integral representing the volume of Solid R. Provide a rationale grounded in integral theory.
(d) Write a definite integral (or sum of definite integrals) that exactly represents the volume of Solid R, taking into account any needed subintervals. Do not evaluate yet.
(e) Evaluate the integral(s) from part (d) to obtain the exact volume of Solid R. Present your work clearly, justify each algebraic manipulation, and state the final answer with units.