7.3 volumes disk and washer methods. suppose i start with this curve. my boss at the acme rocket...
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7.3 Volumes
Disk and Washer Methods
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
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 dx
42
02x
8
2
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
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
Example: Using calculus, derive the formula for finding the volume of a sphere of radius r.
The region bounded by a semicircle and its diameter shown below is revolved about the x-axis, which gives us a sphere of radius r.
dx
Area of each cross section? (circle)2yA What is y?
r–r
Equation of semicircle:222 ryx 22 xry
222 xrA
22 xrA
Example: Using calculus, derive the formula for finding the volume of a sphere of radius r.
Area of each cross section? (circle)
r–r
22 xrA
r
r
dxxrV )( 22
Volume:
(Remember r is just a number!!!)
r
r
xxrV
32
3
1
3232 )(
3
1)(
3
1rrrrrrV
Example: Using calculus, derive the formula for finding the volume of a sphere of radius r.
r–r
Volume:
3232 )(
3
1)(
3
1rrrrrrV
3333
3
1
3
1rrrrV
33
3
2
3
2rrV 3
3
4r
Example: Find the volume of the solid generated when the region bounded by y = x2, x = 2, and y = 0 is rotated about the line x = 2.
dy
Area of each cross section? (circle)
2rA What is r?
r
2
r = 2 – x 22 xA Remember, we are using a dy here!!!So,
xy
xy
2
22 yA
Example: Find the volume of the solid generated when the region bounded by y = x2, x = 2, and y = 0 is rotated about the line x = 2.
Area of each cross section? (circle)
r
2
22 yA Volume:
dyyV )2( 21
Bounds?From 0 to intersection(y-value!!!)
4
0
)2( 21
dyyV 3
8
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
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