Uplift Education / Overview · Web viewVOLUMES of SOLIDS of REVOLUTIONS A solid of revolution is a...
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1 VOLUMES of SOLIDS of REVOLUTIONS A ▪ solid of revolution is a solid that is generated by revolving a plane region about a line that lies in the same plane as the region; the line is called the axis of revolution . DISK METHOD • Find the volume of the solid that is obtained when the region under the curve y=√ x over the interval [1, 4] is revolved about the x-axis. WASHER METHOD dV =πr 2 dx dV =πr 1 2 dx− πr 2 2 dx =π ( r 1 2 −r 2 2 ) dx dV =π [ f ( x) ] 2 dx V = ∫ a b dV =π ∫ a b [ f ( x ) ] 2 dx V = ∫ a b π [ f ( x )] 2 dx =π ∫ 1 4 xdx= π [ x 2 2 ] 1 4 = 15 2 π
Transcript of Uplift Education / Overview · Web viewVOLUMES of SOLIDS of REVOLUTIONS A solid of revolution is a...
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VOLUMES of SOLIDS of REVOLUTIONS
▪ A solid of revolution is a solid that is generated by revolving a plane region about a line that lies in the same plane as the region; the line is called the axis of revolution.
DISK METHOD
• Find the volume of the solid that is obtained when the region under the curve y=√x over the interval [1, 4] is revolved about the x-axis.
WASHER METHOD
dV=π r 2dx
dV=π r12dx−π r2
2dx=π (r12−r 22) dx
dV=π [ f (x ) ]2dx
V=∫a
b
dV=π∫a
b
[ f ( x ) ]2dx
V=∫a
b
π [ f ( x )]2dx=π∫1
4
xdx=π [ x22 ]1
4
=152π
2
• Find the volume of the solid generated when the region between the graphs of the equation
f ( x )=12+ x2 and g ( x )=x over the interval [0, 2] is revolved about the x-axis.
Find the volume of the solid of revolution that is generated by revolving the region R about the x-axis.
“washer-shaped” region with inner radius g(x) and outer radius f(x).
dV=π {[ f ( x ) ]2−[ g ( x ) ]2}dx
V=∫a
b
dV=π∫a
b
{[ f (x ) ]2− [g ( x ) ]2}dx
V=π∫0
2
( [ . 5+x2 ]2−x2 )dx=69π10
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Perpendicular to the y-axis
• The methods or disks and washers have analogs for regions that are revolved about the y-axis. Using the method of slicing, we should have no problem deducing the following formulas for the volumes of the solids.
Disks/Washers
dV=π {[w ( y ) ]2−[ v ( y ) ]2}dy
V=∫c
d
dV=π∫c
d
{[w ( y ) ]2−[v ( y ) ]2}dy
dV=π [u ( y ) ]2dy
V=∫c
d
dV=π∫c
d
[u ( y ) ]2dy
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• Find the volume of the solid generated when the region enclosed by y=√x , y = 2, and x = 0 is revolved about the y-axis.
• Find the volume of the figure generated by rotating the graphs of y = 2x and y = x2 from y = 0 to y = 4 around the line x = 2.
Cylindrical shells method
dV= (2 πr)⏟circumference
[ f (x)]⏟height
dx⏟thickness
V=∫a
b
dV=∫a
b
(2πx) [ f (x)] dx
V=π∫0
2
( y2 )2dy=π y5
5]0
2=32π5
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▪ Find the volume of the solid obtained byrotating about the y-axis the regionbounded by y = 2x2 - x3 and y = 0.
▪ Find the volume of the solid obtained by rotating about the y-axis the regionbetween y = x and y = x2
▪ Find the volume of the solid obtained by rotating the region bounded by y=x−x2 and y=0 about the line x=2
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Volumes of Solids with Known Cross-Sections
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