Greg Kelly, Hanford High School, Richland, Washington
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Transcript of Greg Kelly, Hanford High School, Richland, Washington
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003
Discs
Limerick Nuclear Generating Station, Pottstown, Pennsylvania
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 (disc) is:
2 the thicknessr
In this case:r= the y value of the function
thickness = a small change in x = dx
2x dx
y xThe volume of each flat cylinder (disc) is:
2 the thicknessr
If we add the volumes, we get:
24
0x dx
4
0 x dx
42
02x
8
2x dx
This application of the method of slicing is called the disc 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
Since we will be using the disc method to rotate shapes about other lines besides the x-axis, we will not have this formula on the formula quizzes.
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.
1xy
1 4y
y x
1 12
3
4
1 .7072
1 .5773
12
We use a horizontal disc.
dy
The thickness is dy.
The radius is the x value of the function .1
y
24
1
1 V dyy
volume of disk
4
1
1 dyy
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 disc.
324,700,000 ft
The region bounded by and is revolved about the y-axis.Find the volume.
2y x 2y x
The “disc” 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
2y x
2y x
y x
2y x
2y x
2
24
0 2yV y dy
4 2
0
14
V y y dy
4 2
0
1 4
V y y dy 4
2 3
0
1 12 12
y y
1683
83
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
Like the disc method, this formula will not be on the formula quizzes. I want you to understand the formula.
2y xIf the same region is rotated about the line x=2:
2y x
The outer radius is:
22yR
R
The inner radius is:
2r y
r
2y x
2y x
2y x
y x
4 2 2
0V R r dy
2
24
02 2
2y y dy
24
04 2 4 4
4yy y y dy
24
04 2 4 4
4yy y y dy
14 2 20
13 4 4
y y y dy 43
2 3 2
0
3 1 82 12 3
y y y
16 64243 3
83
Find the volume of the region bounded by , , and revolved about the y-axis.
2 1y x 2x 0y 2 1y x
We can use the washer method if we split it into two parts:
25 2 2
12 1 2 1y dy
21y x 1x y
outerradius
innerradius
thicknessof slice
cylinder
5
14 1 4y dy
5
15 4y dy
52
1
15 42
y y
25 125 5 42 2
25 9 42 2
16 42
8 4 12