Let be the region in the –plane bounded above by the graph of f(x)=|x-1|
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let R be the region in the xy–plane bounded above by the graph of
f(x)=∣x−1∣, for −2≤x≤4,
bounded below by the x–axis, and bounded on the left and right by the vertical lines x=−2 and x=4. The region R is revolved about the horizontal line y=−2 to create a solid of revolution.
(1) Part 1: (a) Determine the x–coordinate at which the analytic expression for f changes. Explain why this point necessitates partitioning the interval [−2,4] when setting up an integral for the volume.
(2) Part 2: (b) Write, but do not yet evaluate, the exact expression (which may be the sum of two definite integrals) representing the volume of the solid obtained by revolving R about y=−2. Clearly indicate the limits of integration for each integral.
(3) Part 3: (c) Evaluate the integral(s) from part (b) to obtain the exact volume of the solid.
(4) Part 4: (d) The graph of f has a corner at x=1. Justify analytically why no additional sub-intervals beyond those identified in part (a) are required to compute the volume accurately.