MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
In the first quadrant, consider the region bounded above by the semicircle x2+y2=9 (with center at the origin), below by the x–axis, and on the left by the y–axis. This region is revolved about the horizontal line y=2 to create a solid.
(a) Express the outer radius R(x) and inner radius r(x) of a representative washer in terms of x, where 0≤x≤3. Simplify each expression.
(b) Write a single definite integral that yields the exact volume of the solid. Do not evaluate the integral.
(c) Without evaluating the integral in part (b), determine whether the volume you wrote is larger than, smaller than, or equal to the volume obtained by revolving the same region about the x–axis. Justify your answer rigorously.
(d) Suppose instead the semicircle had radius R>0. Generalize your work in parts (a) and (b) to write an integral that gives the volume when the corresponding region is revolved about y=2. Clearly specify all intervals and variables used.