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Chapter 6 | Applications of Integration
163. Consider the region enclosed by the graphs of y = f ( x ), y =1+ f ( x ), x =0, y =0, and x = a >0. What is the volume of the solid generated when this region is rotated around the y -axis? Assume that the function is defined over the interval [0, a ]. 164. Consider the function y = f ( x ), which decreases from f (0) = b to f (1) =0. Set up the integrals for determining the volume, using both the shell method and the disk method, of the solid generated when this region, with x =0 and y =0, is rotated around the y -axis. Prove that both methods approximate the same volume. Which method is easier to apply? ( Hint: Since f ( x ) is one- to-one, there exists an inverse f −1 ( y ).)
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