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7.2: Volumes by Slicing
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2001
Little Rock Central High School,Little Rock, Arkansas
Bellwork:Find the area bound by and in the first quadrant.
Area =
The Method of Cross-Sections
Intersect S with a plane Px
perpendicular to the x-axis
Call the cross-sectional area A(x) A(x) will vary as x increases from a to b
Cross-Sections (cont’d) Divide S into “slabs” of equal width
∆xusing planes at x1, x2,…, xn Like slicing a loaf of bread! To add an infinite number of slices of
bread…..we must integrate
b
aV A x dx
The formula can be
applied to any solid for which the
cross-sectional area A(x) can be
found
This includes solids of revolution,
which we will cover today…
…but includes many other solids as
well
A Bigger Picture
b
aV A x dx
Method of Slicing:
1
Find a formula for A(x)dx (OR A(y)dy)
(Note that I used A(x)dx instead of dA(x).)
Sketch the solid and a typical cross section.
2
3 Find the limits of integration.
4 Integrate A(x)dx to find volume. ORIntegrate A(y)dy to find volume.
y x Suppose I start with this curve.
My boss at the ACME Rocket Company has assigned me to build a nose cone in this shape.
So I put a piece of wood in a lathe and turn it to a shape to match the curve.
y xHow could we find the volume of the cone?
One way would be to cut it into a series of thin slices (flat cylinders) and add their volumes.
The volume of each flat cylinder (disk) is:
2 the thicknessr
In this case:
r= the y value of the function
thickness = a small change
in x = dx
2
x dx
𝐴 (𝑥 )𝑑𝑥=¿
y xThe volume of each flat cylinder (disk) is:
2 the thicknessr
If we add the volumes, we get:
24
0x dx
4
0 x dx4
2
02x
8
2
x dx
𝑉=𝑥1
𝑥2
𝐴 (𝑥 ) 𝑑𝑥
=
𝐴 (𝑥 )𝑑𝑥=¿
This application of the method of slicing is called the disk method. The shape of the slice is a disk, so we use the formula for the area of a circle to find the volume of the disk.
If the shape is rotated about the x-axis, then the formula is:
2 b
aV y dx
2 b
aV x dy A shape rotated about the y-axis would be:
The region between the curve , and the
y-axis is revolved about the y-axis. Find the volume.
1x
y 1 4y
y x
1 1
2
3
4
1.707
2
1.577
3
1
2
We use a horizontal disk.
dy
The thickness is dy.The radius is the x value of the function .1
y
24
1
1 V dy
y
volume of disk
4
1
1 dy
y
4
1ln y ln 4 ln1
02ln 2 2 ln 2
𝐴 (𝑦 )𝑑𝑦=𝜋 (1/√ 𝑦 ) 2
The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis:
2.000574 .439 185x y y x
y
500 ft
500 22
0.000574 .439 185 y y dy
The volume can be calculated using the disk method with a horizontal disk.
324,700,000 ft
The region bounded by and is revolved about the y-axis.Find the volume.
2y x 2y x
The “disk” now has a hole in it, making it a “washer”.
If we use a horizontal slice:
The volume of the washer is: 2 2 thicknessR r
2 2R r dy
outerradius
innerradius
2y x
2
yx
2y x
y x
2y x
2y x
2
24
0 2
yV y dy
4 2
0
1
4V y y dy
4 2
0
1
4V y y dy
42 3
0
1 1
2 12y y
168
3
8
3
𝐴 (𝑦 )𝑑𝑦=¿
This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle.
The washer method formula is: 2 2 b
aV R r dx
2y xIf the same region is rotated about the line x=2:
2y x
The outer radius is:
22
yR
R
The inner radius is:
2r y
r
2y x
2
yx
2y x
y x
4 2 2
0V R r dy
2
24
02 2
2
yy dy
24
04 2 4 4
4
yy y y dy
24
04 2 4 4
4
yy y y dy
14 2 2
0
13 4
4y y y dy
432 3 2
0
3 1 8
2 12 3y y y
16 64
243 3
8
3
p
Second Example of Washers
Problem Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves y = x andy = x2
…but about the line y = 2 instead of thex-axis
The solid and a cross-section are illustrated on the next slide
Second Example of Washers (cont’d)
Second Example of Washers (cont’d)
Solution Here
So
2 22 4 22 2 5 4A x x x x x x
1 1 4 2
0 0
15 3 2
0
5 4
85 4
5 3 2 15
V A x dx x x x dx
x x x
The formula can be
applied to any solid for which the
cross-sectional area A(x) can be
found
This includes solids of revolution, as
shown above…
…but includes many other solids as
well
A Bigger Picture
b
aV A x dx
The Method of Cross-Sections
Intersect S with a plane Px
perpendicular to the x-axis
Call the cross-sectional area A(x) A(x) will vary as x increases from a to b
Cross-Sections (cont’d) Divide S into “slabs” of equal width
∆xusing planes at x1, x2,…, xn Like slicing a loaf of bread! To add an infinite number of slices of
bread…..we must integrate