MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Region S is enclosed by the graphs of y=x2 and y=2x for 0≤x≤2. For each x in this interval, a rectangular cross-section perpendicular to the x–axis is constructed whose height is three times the length of its base (the base spans the region from the lower curve to the upper curve at that x-value).
(a) Set up, but do not yet evaluate, an expression for the volume of the solid obtained in this manner.
(b) Evaluate the integral from part (a) to obtain the exact volume of the solid.
(c) Suppose instead that the height of each cross-section were k times its base, where k>0. Write a formula for the volume in terms of k and determine the value of k that would double the volume found in part (b).