Section6_2_review

Section6_2_review - Section 6.2: Volumes The volume of a...

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Section 6.2: Volumes The volume of a solid of revolution is obtained by evaluating: V = Ax ( 29 dx a b , if the solid is obtained by rotating a plane region about the x -axis or a line parallel to the x -axis, where a , b [ ] corresponds to variable x . V = A ( y ) dy c d , if the solid is obtained by rotating a plane region about the y -axis or a line parallel to the y -axis, where c , d [ ] corresponds to variable y . The cross-sectional area, Ax ( 29 or Ay ( 29 , is obtained as follows: If the cross-section is a disk, then obtain the radius of the disk (as either a function of x or a function of y , whichever is appropriate), and use A radius ( 29 2 . If the cross-section is a washer, then obtain the inner radius and the outer radius of the washer (as either a function of x or a function of y , whichever is appropriate), and use A ou ter rad ius ( 29 2 - inner rad ius ( 29 2 [ ] . Any radius that is required is always measured from the axis of revolution. Example. Find the volume of the solid obtained when the region bounded by the x -axis, the y -axis, and the line y = x + 3 is rotated about the x -axis. Solution. The region is sketched below. Since the axis of revolution is the x -axis, slice perpendicular to the x -axis and obtain a piece with width x . Since x becomes dx in the definition of the definite integral, we need to evaluate V = Ax ( 29 dx a b , where a , b [ ] =- 3 ,0 [ ] . In other words, we need the function for area expressed in terms of x . Because the plane region lies directly against the axis of revolution, the

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This note was uploaded on 01/12/2010 for the course MATH 44245 taught by Professor Famiglietti during the Fall '07 term at UC Irvine.

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Section6_2_review - Section 6.2: Volumes The volume of a...

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