Consider the region in the plane that is enclosed by the curves and for
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Consider the region S in the plane that is enclosed by the curves x=y2 and x=2y for 0≤y≤4. The region is revolved about the vertical line x=−1.
(a) For integration with respect to y, write an integral that computes the exact volume of the resulting solid. Clearly label inner and outer radii in terms of y before writing the integral.
(b) Rewrite each bounding curve to express y explicitly as a function of x, and determine the x–values that define the region.
(c) Using the results from part (b), write an integral with respect to x that computes the same volume.
(d) Evaluate one of the integrals to obtain a numerical value for the volume, and verify that substituting the bounds into the other integral leads to the same result.