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Unformatted text preview: Math 28 Dr. C. Famiglietti ?kfrrr***** rr * * ?b* * * *** rs t(** ***:l* tc*f.*** * ***** * tr ttrb?trt* r( rr *tr*tr* rr?k*** * ***** t<:k:b*** 'r 'k Section 5.2: Volumes A solid of revolution is a solid obtained by rotating a plane region about a line. The line that the solid of rotated about is called the axis of revolution The volume of a solid of revolution is obtained by evaluating: b V = ! e(x)dx, if thesolid is obtained by rotating a plane region about the xaxis or a line parallel to the xaxis, where a < x < b  d , = {Orrrdy,if the solid is obtained by rotating aplane region about theyaxis or a line parallel to theyaxis, where c s y s d. For any given problem, the appropriate integral (with respect to x or with respectto y) can be obtained by slicing perpendicular to the axis of revolution and determining whether the slice has thickness Ax or ff . If the slice is Ax thick, then since Ax becomes dx,youshould evaluate V = [ l(x)dx. However if the slice is Ay thick, then since Ay becomes dy,youshould evaluate V =i e1y1ay. c The integrand of the above integrals is the crosssectional area, A(.r) or A(y), obtained as follows: . If the crosssection is a disk (the axis of revolution lies directly against the plane region), then obtain the radius of the disk (as either a function ofr or a function of y, whichever is appropriate), and use A = n(radiur)'= or"  . If the crosssection is a washer (there is empty space between the axis of revolution and the plane region), ttren obtain the inner radius and the outer radius of the washer (as either a function ofx or a function ofy, whichever is appropriate), and use A : a[(outer radius)'  (inner radius)'] : ,({r*,)'  (t,)t) The radius, whether itbe for a disk or a washer, is always measured from the axis of revolution and perpendicular to the axis of revolution. 5'x+V Example. Find the volume of the solid obtained when the region bounded by the xaxis, theyaxis, and the line y = x * 3 is rotated about the raxis.the raxis....
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This note was uploaded on 06/07/2008 for the course MATH Math2b taught by Professor Famigleitti during the Spring '08 term at UC Irvine.
 Spring '08
 Famigleitti
 Math, Calculus

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