MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Region T is bounded above by the line y=3−2x, below by the curve y=ln(x+2), and above by the x-axis. The two bounding curves intersect exactly once for x>−2. The region T is revolved about the x-axis to produce a solid.
(a) Find, to three decimal places, the x-coordinates where y=3−2x intersects y=ln(x+2).
(b) Using the values from part (a), specify the x-intervals on which each curve serves as the upper boundary of T.
(c) Derive the definite integral(s) that together compute the volume of the solid generated by revolving T about the x-axis.
(d) Evaluate the integral expression(s) numerically to three decimal places.
(e) Compute the individual volumes contributed by each sub-region identified in part (b) and determine which sub-region accounts for the greatest portion of the total volume, justifying your conclusion with numerical evidence.