x = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x " 0 :
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1 Clculus II MAT 146 Integrtion Applictions: Volumes of 3D Solids Our gol is to determine volumes of vrious shpes. Some of the shpes re the result of rotting curve out n xis nd other shpes re simply given s 3-dimensionl ojects. The generl process we use in mny prolem situtions is to determine the volume of some typicl slice of the solid nd then use integrtion to sum the volumes of n infinite numer of such typicl slices. The volume of typicl slice, in turn, cn often e pproximted y clculting the re of fce of the slice nd multiplying tht re y the thickness of the slice. ( ) hs een Here, the curve y = f x rotted out the x-xis from x = to x =. A slice of the resulting 3- dimensionl solid hs een sketched in. Wht is the re of the fce of tht typicl slice? Wht is the volume of tht typicl slice? x = typicl slice y = f(x) x = prt of the curve y = f(x) rdius: f(x) x-xis (xis of rottion) The fce of the typicl slice is circle. Its rdius extends from the xis of rottion to the curve itself. Therefore, this typicl slice must hve rdius equl to f ( x), the vlue of the function t x. thickness of the slice: Δx To determine the volume of the solid, we use definite integrl to sum the volumes of the slices s we let x " : Volume = ( fce re)(slice thickness) = (re of circle)(su-division size) = rdius ( ) dx = ( ( )) dx = f (x) dx f x This technique for clculting the volume of solid of rottion is often clled the disk method ecuse typicl slice is disk.
2 Exmple #1: Determine the volume of the solid of revolution creted when the region ounded y y = x, y =, nd x = is rotted out the x-xis. Step 1: Drw picture of the region to e rotted nd picture of the rottion imge. Include n illustrtion of typicl slice. Sketch the oundries nd identify the region to e rotted. Reflect the region out the xis of rottion. Visulize the rottion y representing third dimension, including sketch of typicl slice. Step : Isolte typicl slice nd clculte its volume. Volume of slice = ( fce re)(slice thickness) = (re of circle)(slice thickness) = ( rdius) x = ( x ) x = x 4 x
3 Step 3: Set up definite integrl to represent the volume of the solid of rottion. Totl Volume = (volume of typicl slice) = ( fce re)(slice thickness) = rdius Step 4: Evlute the definite integrl. ( ) dx = " %( Totl Volume = x 4 dx = $ x5 '* # 5 &) x ( ) dx = x 4 dx = 3 cuic units 5 Exmple #: Using the sme region s for Exmple 1, determine the volume of the solid of revolution creted when the region is rotted out the line y = 1. Step 1: Drw picture nd illustrte typicl slice. Sketch the oundries nd identify the region to e rotted. Reflect the region out the xis of rottion. Visulize the rottion y representing third dimension, including sketch of typicl slice.
4 Step : Isolte typicl slice nd clculte its volume. Here, typicl slice is not solid disk ut looks like wsher, disk with hole in the middle. Illustrted here, this method is therefore clled the wsher method for determining the volume of solid of revolution. Volume of slice = ( fce re)(slice thickness) = (re of outside circle re of inside circle)(slice thickness) ( ) $ % " ( inside rdius) $ { # %} &x ( ) $ % " ( r ) $ { # %} &x = " # outside rdius = " # R ( ) $ %( " ( 1 ) $ { # %} &x " = 1+ x #' Step 3: Set up definite integrl to represent the volume of the solid of rottion. Totl Volume = (volume of typicl slice) = ( fce re)(slice thickness) ( ) % &' ( " ( 1 ) % { # &} dx " = 1+ x #$ Step 4: Evlute the definite integrl. Totl Volume = ( 1+ x "# ( ) $ %& ' ( 1 ) $ { " %} dx ( {( ) '1 } dx = 1+ x = 176 cuic units 15 Exmple #3: Using the sme region s for Exmple 1, determine the volume of the solid of revolution creted when the region is rotted out the y-xis. This rottion genertes owl-like solid. We su-divide the x-xis intervl from x = to x = into su-intervls of size x. This cretes sequence of shells, ech similr to piece of tuing, sy, from pper towel roll. We
5 unwrp (fltten) ech shell to get three-dimensionl solid whose volume is the product of length, width, nd height. Step 1: Drw picture nd illustrte typicl shell. Sketch the oundries nd identify the region to e rotted. Reflect the region out the xis of rottion. Here, tht s the y-xis. Visulize the rottion y representing third dimension, including sketch of typicl shell. Step : Isolte typicl shell nd clculte its volume. Here is typicl shell, with its length, width, nd height identifed. This method is clled the shell method for determining the volume of solid of revolution. Volume of shell = (shell circumference)(thickness)(height) = (r)(x)( f (x)) ( ) = ( x)(x) x = ( x) ( x )x
6 Step 3: Set up definite integrl to represent the volume of the solid of rottion. Totl Volume = (volume of typicl shell) = (shell circumference)(thickness)(height) = ( x) ( x )dx Step 4: Evlute the definite integrl. Totl Volume = ( x) ( x )dx = ( x 3 )dx " = x 4 %( $ '* = 8 cuic units # 4 &)
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