A planar region consists of two adjoining parts: • Region is bounded by the x-axis…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A planar region consists of two adjoining parts:
• Region A is bounded by the x-axis and the line y=2−x for 0≤x≤1.
• Region B is bounded by the x-axis and the parabola y=x2 for 1≤x≤2.
The combined region A∪B is revolved about the x-axis to create a solid T.
(a) Justify analytically why two separate definite integrals are required to express the volume of T rather than a single integral over 0≤x≤2. Your justification should reference the behavior of the bounding functions on each subinterval.
(b) Set up, but do not evaluate, the complete expression (which may be written as a sum of definite integrals) that gives the volume of T.
(c) Evaluate each integral in the expression from part (b) to obtain the exact volume of T.