aim: how do we find volume using the shell method?
DESCRIPTION
Aim: How do we find volume using the Shell Method?. Do Now:. Find the volume of the solid that results when the region bounded by x = y 3 and x = y 2 from y = 0 to y = 1 is revolved about the y -axis. h. h. w. Axis of Revolution. p. p. The Shell Revolution. h. w. - PowerPoint PPT PresentationTRANSCRIPT
Aim: Shell Method for Finding Volume Course: Calculus
Do Now:
Aim: How do we find volume using the Shell Method?
Find the volume of the solid that results when the region bounded by x = y3 and x = y2 from y = 0 to y = 1 is revolved about the y-axis.
Aim: Shell Method for Finding Volume Course: Calculus
The Shell Revolution
w
h
p
2
w
2
w
2
w
2
w
p
2
wp
2
wp
2
wp
2
wp
Axis of Revolution
h
Aim: Shell Method for Finding Volume Course: Calculus
The Shell Method
w
h
2
wp
2
wp
2
wp Outer radius
2
wp Inner radius
(volume of cylinder)
Volume of shell = minus ( )
(volume of hole)
Aim: Shell Method for Finding Volume Course: Calculus
The Shell Method
(volume of cylinder)
Volume of shell = minus ( )
(volume of hole)
1
2n
i ii
V p y h y y
2d
cV p y h y dy
take to limit
2 2
2 2
2
2 average radius height thickness
w wV p h p h
phw
outer radius
inner radius
Aim: Shell Method for Finding Volume Course: Calculus
To find the volume of a solid of revolution with the shell method
The Shell Method
Horizontal Axis of Revolution
2d
cV p y h y dy
c
d
Horizontal Axis of Revolution
( )p y
y
h(y)
Vertical Axis of Revolution
2b
aV p x h x dx
Vertical Axis of Revolution
x
h(x)
a b
p(x)
Aim: Shell Method for Finding Volume Course: Calculus
Model Problem
Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x3 and the x-axis (0 < x < 1) about the y-axis.
0.4
0.2
0.5 1
f x = x-x3
p(x) = x
h(x) = x – x3
2b
aV p x h x dx Vertical Axis of Revolution
1 3
02V x x x dx
15 31 4 2
00
2 25 3
x xV x x dx
1 1 42 0 0
5 3 15V
Aim: Shell Method for Finding Volume Course: Calculus
Model Problem
Find the volume of the solid of revolution formed by revolving the region bounded by and the y-axis (0 < y < 1) about the x-axis.
2yx e
1
0.5
1
f y = e -y2
h(y)2ye
p(y)
Horizontal Axis of Revolution
2d
cV p y h y dy
2 1
0
11 1.986
ye
e
21
02 yV ye dy
Aim: Shell Method for Finding Volume Course: Calculus
Comparing Disc and Shell Methods
Disc Method
Shell Method
3
2
1
-1
2
x
Rc
d
r
3
2
1
-1
2
ab
2b
aV phdx
x
p
h
3
2
1
-1
2
y
c
d
p
h
2d
cV phdy
3
2
1
-1
2a bR
ry
2 2d
cV R r dy 2 2b
aV R r dx
Aim: Shell Method for Finding Volume Course: Calculus
Do Now:
Aim: How do we find volume using the Shell Method?
The bases of a solid is the region enclosed by the graphs of y = 1/2x2 and y = 8. Cross sections perpendicular to the y-axis are semicircles with diameter in the plane of the region. Find the volume of the solid.
10
8
6
4
2
-5 5
g x = 8
f x = 0.5x2
diameter
Aim: Shell Method for Finding Volume Course: Calculus
Do-Now
212 radius of semi-cirlces
2y x x y
21
2A r
212
2A y
Volume = ( )b
aA y dy
28
0
12
2y dy
32 cubic units
Aim: Shell Method for Finding Volume Course: Calculus
-1 1
2.5
2
1.5
1
0.5
A: (1, 2)
f x = x2+1
Recall: Model Problem
2
Find the volume of the solid formed
by revolving the region bounded by
the graphs of 1, 0, 0,
and 1 about the -axis.
y x y x
x y
y
y
r
R
1 2 2
0
22 2
1
1 0
1 1
V dy
y dy
disk
washer
1 2
0 11 2V dy y dy
22
1
01
22
1 34 2 2
2 2
yy y
disk
washer
0, 0 1( )
1, 1 2
yr y
y y
1R
Aim: Shell Method for Finding Volume Course: Calculus
-1 1
2.5
2
1.5
1
0.5
A: (1, 2)
f x = x2+1
Shell Method is Preferable
2
Find the volume of the solid formed
by revolving the region bounded by
the graphs of 1, 0, 0,
and 1 about the -axis.
y x y x
x y
p(x)
h(x) = x2 + 1
= x
2b
aV p x h x dx Vertical Axis of Revolution
1 2
02 1V x x dx
14 2
0
24 2
3 32
4 2
x x
Aim: Shell Method for Finding Volume Course: Calculus
2
1
-1
-2
-5 5
f x = 1-x2
16
Model Problem
A pontoon is to be made and is designed by rotating the graph of
about the x-axis, where x and y are measured in feet. Find the volume.
2
1 , 4 416
xy x
8 ft.
2 ft.
Disk method a. rev
Shell method a. rev
Aim: Shell Method for Finding Volume Course: Calculus
A pontoon is to be made and is designed by rotating the graph of
about the x-axis, where x and y are measured in feet. Find the volume.
2
1
-1
-2
-5 5
f x = 1-x2
16
Model Problem – Disk Method
2
1 , 4 416
xy x
2 2( ) ( )
b
aV R x r x dx
R(x) = 2
116
x
r(x) = 0
224
41
16
xV dx
2 44
41
8 256
x xdx
43 5
4
64 cubic feet
24 1280 15
x xx
Aim: Shell Method for Finding Volume Course: Calculus
A pontoon is to be made and is designed by rotating the graph of
about the x-axis, where x and y are measured in feet. Find the volume.
2
1
-1
-2
-5 5
f x = 1-x2
16
Model Problem – Shell Method
2
1 , 4 416
xy x
p(y) = y
h(y)
4 1x y
Horizontal Axis of Revolution
2d
cV p y h y dy
1
02 8 1V y y dy
8 1 y
2( ) = h(y)
64 cubic feet
15
Aim: Shell Method for Finding Volume Course: Calculus
3
2.5
2
1.5
1
0.5
-1 1 2 3
h(x) = x3 + x + 1
g y = 2
Model Problem – Only Shell
Find the volume of the solid of revolution formed by revolving the region bounded by the graphs of y = x3 + x + 1, y = 1, x = 1 about the line x = 2.
Axis of Revolution
= 2 – x p(x)
(1, 3)
- ( 1)
can’t solve for x
Aim: Shell Method for Finding Volume Course: Calculus
3
2.5
2
1.5
1
0.5
-1 1 2 3
h(x) = x3 + x + 1
g y = 2
Model Problem – Only ShellFind the volume of the solid of revolution formed by revolving the region bounded by the graphs of y = x3 + x + 1, y = 1, x = 1 about the line x = 2.
Axis of Revolution
p(x)= 2 – x
(1, 3)
- ( 1)
2b
aV p x h x dx Vertical Axis of Revolution
1 3
02 2 1 1V x x x dx
1 4 3 2
02 2 2x x x x dx
15 4 32 2
0
1 1 1 292 2 1
5 2 3 5 2 3 15
x x xx
can’t solve for x
Aim: Shell Method for Finding Volume Course: Calculus
The Shell Method