Consider the region in the -plane bounded by the graph of and the -axis for
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Consider the region S in the xy-plane bounded by the graph of y=∣x−1∣+1 and the x-axis for 0≤x≤2. A solid is formed by taking cross-sections perpendicular to the x-axis; each cross-section is an equilateral triangle whose base lies in S at that x-value.
Answer all parts completely, showing every algebraic step.
(a) Rewrite the defining equation of the boundary curve as an explicit piecewise function of x on the interval 0≤x≤2, identifying the point(s) at which the expression changes form.
(b) Based on part (a), set up—but do not yet evaluate—the definite integral(s) required to compute the volume of the solid. Make sure to account for every sub-interval where the integrand changes form.
(c) Evaluate the integral(s) from part (b) to obtain the exact volume of the solid. Present your final answer in simplest exact form.