19fa-1910-recitation06-solutions.pdf - \u00a76.3 V OLUMES OF REVOLUTION \u00a76.4 C YLINDRICAL SHELLS N AME S OLUTIONS Math 1910 T HE DISK WASHER METHOD(1 The

19fa-1910-recitation06-solutions.pdf - u00a76.3 V OLUMES...

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§ 6.3 V OLUMES OF REVOLUTION § 6.4 C YLINDRICAL SHELLS N AME : S OLUTIONS Math 1910 T HE DISK / WASHER METHOD (1) The Disk Method: If f ( x ) 0 on [ a , b ] , then the solid obtained by rotating the region under the graph around the x -axis has volume Z b a πf ( x ) 2 dx ( 1 ) . (2) The Washer Method: If f ( x ) g ( x ) 0 on [ a , b ] , then the solid obtained by rotating the region between f ( x ) and g ( x ) around the x -axis has volume Z b a π ( f ( x ) 2 - g ( x ) 2 ) dx ( 2 ) . T HE SHELL METHOD (3) Shell Method : When you rotate the region between two graphs around an axis, the segments parallel to the axis generate cylindrical shells. The volume V of the solid of revolution is the integral of the surface areas of the shells. V = Z ( radius )( height of shell ) dr ( 3 ) . 1
PROBLEMS(1)Sketch the region enclosed by the curves, and determine the cross section perpendicular to thex-axis.Set up an integral for the volume of revolution obtained by rotating the region around thex-axis, butdo not evaluate. 2

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