Solid of Revolution
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Transcript of Solid of Revolution
Solid of Revolution
Revolution about x-axis
What is a Solid of Revolution - 1Consider the area under the graph of y = 0.5x from x = 0 to x = 1:
What is a Solid of Revolution - 2If the shaded area is now rotated about the x-axis, then a three-dimensional solid (called Solid of Revolution) will be formed:
http://chuwm2.tripod.com/revolution/Pictures from
What will it look like?
What is a Solid of Revolution - 3 Actually, if the shaded triangle is regarded as made up of straight lines perpendicular to the x-axis, then each of them will give a circular plate when rotated about the x-axis. The collection of all such plates then pile up to form the solid of revolution, which is a cone in this case.
Finding Volume
http://clem.mscd.edu/~talmanl/HTML/VolumeOfRevolution.html
How is it calculated - 1Consider the solid of revolution formed by the graph of y = x2 from x = 0 to x = 2:
What will it look like?
http://www.worldofgramophones.com/victor-victrola-gramophone-II.jpg
How is it calculated - 2Just like the area under a continuous curve can be
approximated by a series of narrow rectangles, the volume of a solid of revolution can be approximated by a series of thin circular discs:
we could improve our accuracy by using a larger and larger number of circular discs,making them thinner and thinner
How is it calculated - 3x
x
x
As n tends to infinity,It means the discs get thinner and thinner.
And it becomes a better and better approximation.
As n tends to infinity,It means the discs get thinner and thinner.
And it becomes a better and better approximation.It can be replaced by an integral
Volume of Revolution Formula
The volume of revolution about the x-axis between x = a and x = b, as , is :
This formula you do need to knowThink of is as the um of lots of circles
… where area of circle = r2
Δx = 0
limΔx→ 0
π f x( )2⎡⎣ ⎤⎦x=a
x=b
∑ Δx = π f x( )2⎡⎣ ⎤⎦a
b
x
0
1
2
1 2 3 4
y = x How could we find the volume of the cone?
One way would be to cut it into a series of disks
(flat circular cylinders) and add their volumes.
The volume of each disk is:2 the thicknessrπ ⋅
In this case:r= the y value of the function
thickness = a small change in x = dx
π π x( )
2
x
→
Example of a disk
0
1
2
1 2 3 4
y = xThe volume of each flat cylinder (disk) is:
2 the thicknessrπ ⋅
If we add the volumes, we get:
( )24
0x dxπ∫
4
0 x dxπ=
42
02xπ
= 8π=
π x( )
2
x
Example 1
dxx1
0
2)5.0(
Consider the area under the graph of y = 0.5x from x = 0 to x = 1:
What is the volume of revolution about the x-axis?
Integrating and substituting gives:
0.5 1
π y2
a
b
∫ dx π (0.5x)2
0
1
∫ dx = π4
x2
0
1
∫ dx
π4
x3
3⎡⎣⎢
⎤⎦⎥0
1
=π4
13
− 0⎡⎣⎢
⎤⎦⎥
=π12
Example 2
between x = 1 and x = 4
What is the volume of revolution about the x-axis
Integrating gives:
xx1y +
=for
π y2
a
b
∫ dx π1 + x
x⎛⎝⎜
⎞⎠⎟
2
1
4
∫ dx = π1 + x
x1
4
∫ dx
=π1x1
4
∫ +xx
dx = π1x1
4
∫ +1dx
π ln x + x⎡⎣ ⎤⎦1
4= π ln 4 + 4 − ln1−1[ ] = π (ln 4 + 3)
http://clem.mscd.edu/~talmanl/HTML/DetailedVolRev.html
Sphere Torus
x
y
x
y
What would be these Solids of Revolution about the x-axis?
Sphere Torus
x
y
x
y
What would be these Solids of Revolution about the x-axis?
• http://curvebank.calstatela.edu/volrev/volrev.htm