Download - 6049 Volume AP Calculus. Volume Volume = the sum of the quantities in each layer where h is # layers
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6049 Volume
AP Calculus
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Volume
Volume = the sum of the quantities in each layer
πβπ€βh
where h is # layers
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x
x
y
xx
x-axis
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Volume by Cross Sections
n
( )x thickness h
foundation
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BE7250
Axial (cross sectional) magnetic resonance image of a brain with a large region of acute infarction, formation of dying or dead tissue, with bleeding. This infarct involves the middle and posterior cerebral artery territories.Credit: Neil Borden / Photo Researchers, Inc.
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Volume by Slicing(Finding the volume of a solid built on the base in the x β y plane)
METHOD:
1.) Graph the βBASEβ
2.) Sketch the line segment across the base.
That is the representative slice βnβ
Use βnβ to find: a.) x or y (Perpendicular to axis)
b.) the length of βnβ
3.) Sketch the βCross Sectional Regionβ - the shape of the slice (in 3-D )
from Geometry V = B*h h = or is the thickness of the slice
B = the Area of the cross section
4.) Find the area of the region
5) Write a Riemannβs Sum for the total Volume of all the Regions
β π₯ ππππππ’
πππππππ
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Example 1: The base of a solid is the region in the x-y plane bounded by the graph
and the y β axis. Find the volume of the solid if every cross section by a plane perpendicular to the x-axis is a square.
2 3x y
Base
Cross -Section
x
y
π¦ 2=3 βπ₯π¦=Β±β3 βπ₯
ππ= limπβ β
β0
3
(2β3 βπ₯ )2 β π₯
β«0
3
(2β3 βπ₯ )2ππ₯
π¦=β3 βπ₯
π¦=ββ3β π₯
h
π=(β3βπ₯ ) β (ββ3βπ₯ )
π=2β3 βπ₯π=π΅βhπ=π2 βπ₯π= (2β3βπ₯ )2
β π₯
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Example 2: The base of a solid is the region in the x-y plane bounded by the graph
and the y β axis. Find the volume of the solid if every cross section by a plane perpendicular to the x-axis is an Isosceles Rt. Triangle (leg on the base).
2 3x y
x
y
nn π=π΅βh
π=12
(π2 ) β π₯
π¦=β3 βπ₯
π¦=ββ3β π₯
h
π=(β3βπ₯ ) β (ββ3βπ₯ )
π=2β3 βπ₯
ππ= limπβ β
β0
3
( 12
(( 2β3β π₯ )2 ))βπ₯
β«0
3
( 12
( (2β3βπ₯ )2 ))ππ₯
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Some Important Area Formulas
Square-
side on base
Square-
diagonal on base
Isosceles rt Ξ
leg on base
Isosceles rt Ξ
hypotenuse on base
Equilateral Ξ
Semi - Circle Circle
diameter on the base
π΅=π2
n
nnn
n
n
n
π΅=π2
2
π΅=12π2
π΅=π2
4
π2
β3π΅=1
2π(π2 )β3
π΅=π2
4β3
π΅=π π2
4π΅=1
2 (π2 )2
π
π΅=π π2
8
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EXAMPLE #4/406The solid lies between planes perpendicular to the x - axis at x = -1 and x = 1 . The cross sections perpendicular to the x β axis are circular disks whose diameters run from the parabola y = x2 to the parabola y = 2 β x2.
π₯2=2 βπ₯2
2 π₯2=2π₯2=1π₯Β±β1
π=β π₯ [β 1,1 ]π=(2βπ₯2 ) β (π₯2 )π=2 β2 π₯2n
π=π΅βh
π= ππ2
4β π₯
π=π4
(2 β2 π₯2 )2 βπ₯
ππ= limπβ β
β π4
(2 β2π₯2 )2β π₯
ππ=β« π4
(2 β2 π₯2)2ππ₯
ππ=π4 β« ( 4 β 8π₯2+4 π₯4 )ππ₯
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The base of a solid is the region bounded by The cross sections, perpendicular to the x-axis, are rectangles whose height is 3x
4=π₯2
Β± 2=π₯
h=βπ₯ [β 2,2 ]π=4 βπ₯2
π=π΅βhπ= [ ( 4βπ₯2 ) 3π₯ ] β π₯
limπβ β
β [ (12 π₯β 3π₯3 ) ] βπ₯
β« [ (12π₯β3 π₯3 ) ]ππ₯
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Assignment:
P. 406 # 1 - 6 all If the problem has multiple parts work β a β then set up only ( to the definite integral) the other parts.
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Volumes of Revolution:
Disk and Washer Method
AP Calculus
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Volume of Revolution: Method Lengths of Segments:In revolving solids about a line, the lengths of several segments are needed for the radii of disks, washers, and for the heights of cylinders.A). DISKS AND WASHERS 1) Shade the region in the first quadrant (to be rotated) 2) Indicate the line the region is to be revolved about.
3) Sketch the solid when the region is rotated about the indicated line.4) Draw the representative radii, its disk or washer and give their lengths.<<REM: Length must be positive! Top β Bottom or Right β Left >>
Ro = outer radius
ri = inner radius
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Disk MethodRotate the region bounded by f(x) = 4 β x2 in the first quadrant about the y - axis
The region is _______________ _______ the axis of rotation.
The Formula: The formula is based on the
_____________________________________________
adjacent to
Volume of a cylinder
π=π π2 h
h=β π¦ [ 0,4 ]π=β4 βπ¦β 0
Right-left
π¦=4 βπ₯2
4 β π¦=π₯2
Β±β4 βπ¦=π₯
πβ« (β4 βπ¦ )2ππ¦
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Washer MethodRotate the region bounded by f(x) = x2, x = 2 , and y = 0 about the y - axis
The region is _______________ __________ the axis of rotation.
The Formula: The formula is based on
_____________________________________________
Separate from
Big cylinder-small cylinder
π=π π β2 hβππ 2h
π (π 2 βπ2 ) h
limπβ β
β π ( (2 )2β (βπ¦ )2 )β π¦
h=β π¦ [ 0,4 ]π =2 β0π=βπ¦β 0
πβ«0
4
( 4 β π¦ )ππ¦
π=8π
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Disk MethodRotate the region bounded by f(x) = 2x β 2 , x = 4 , and y = 0 about the line x = 4
The region is _______________ _______ the axis of rotation.Adjacent to
π·ππ ππ π2 h π¦=2 π₯β2
π₯=π¦+2
2β π¦ [ 0,6 ]
π=4 β( π¦+22 )
π=4 β12π¦β 1
π=(3 β12π¦ )
π=13ππ2 h
π=13π (3 )2 (6 )
π=18π
πβ«0
6
(3 β12π¦ )
2
ππ¦
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Washer Method
Rotate the region bounded by f(x) = -2x + 10 , x = 2 , and y = 0 about the y - axis
The region is _______________ __________ the axis of rotation.Separate from
π¦=β 2π₯+10
π₯=5β12π¦
π =5 β12π¦
π=2
h=β π¦ [ 0,6 ]π (π 2 βπ2 ) h
limπβ β
β ((5 β12π¦ )
2
β (2 )2)β π¦
V=
Separate
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Example 1:
The region is bounded by Rotated about:
the x-axis, and the y-axis a) The x-axis
b) The y-axis
c) x = 3
d) y = 4
24y x
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Example 2:
The region is bounded by: Rotated about:
f(x) = x and g(x) = x2 a) the x-axis in the first quadrant b) the y-axis
c) x = 2d) y = 2
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Example 2: The base of a solid is the region in the x-y plane bounded by the graph
and the x β axis. Find the volume of the solid if every cross section by
a plane perpendicular to the x-axis is an Isosceles Rt. Triangle (leg on the base).
2 3x y