A region in the first quadrant is bounded below by the x-axis
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A region R in the first quadrant is bounded below by the x-axis. For 0≤x≤1 the region is bounded above by the line y=2x, and for 1≤x≤3 it is bounded above by the line y=4−x. The region is revolved about the horizontal line y=−1 to generate a solid S.
(a) Verify that the two upper boundary curves intersect at the point (1,2). Show that the region R may be written as the union R=R1∪R2, where R1 corresponds to 0≤x≤1 and R2 corresponds to 1≤x≤3.
(b) For each subregion R1 and R2, write a definite integral with respect to x whose value equals the volume of the part of S generated by revolving that subregion. Do not combine or evaluate the integrals in this part.
(c) Evaluate the integrals from part (b) to obtain the exact volume of S.
(d) State the units of the resulting volume and explain how the limits of integration reflect the geometric change in the outer radius at x=1.