MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let U be the region bounded by the graph of y=4, the graph of y=∣x∣, and the vertical lines x=−3 and x=3. The region U is revolved about the horizontal line y=−3 to form a solid.
(a) Explain why the expression for the radius of a typical disc must be split into two cases or written with an absolute-value symbol when setting up the integral for the volume.
(b) Write a piecewise-defined function R(x) for the radius of the disc as a function of x on the interval [−3,3].
(c) Set up a single definite integral, using the piecewise definition from part (b) or absolute values, that represents the volume of the solid.
(d) Evaluate the integral to obtain the exact volume.