Math 1B
Overview: Some volume problems are difficult to handle using disks and washers. Consider the following:
Find the volume of the solid obtained by revolving the region about the y -axis.
Integrating cross sectional areas would be difficult since the cross sections are of 4 different types.
The method of cylindrical shells will make this problem easier.
To describe this alternative, recall the following:
Let S be the solid generated by revolving the region R about the y -axis.
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If we divide [ ] into n subintervals of equal width subinterval, we get
Δ x and let x i
* be the midpoint of the i th
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Adding the volumes of each shell, we get an approximation for the total volume. n
Volume ≈
2 π x i
* ( ) Δ x i = 1
The approximation gets better as the shells get thinner (as n
( ) Δ x n
→ ∞
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Volume = lim n →∞
2 π x i
* i = 1
But we recognize the right side as a definite integral. Therefore,
), so
€ b
Volume = 2 π ( ) dx a
The best way to remember and use this formula (and its variations) is as follows:
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Same for revolving about the x -axis:
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Note: There are a couple of important differences between this method and the method of disks/washers that we should note before moving on. First, rotation about a vertical axis will give an area that is a function of x and rotation about a horizontal axis will give an area that is a function of y .
This is exactly opposite of the method of disks/washers.
Second, we don’t take the complete range of x or y for the limits of integration as we did in the previous section. Instead we take a range of x or y that will cover one side of the solid. If we expand out the radius to cover one side we will automatically expand in the other direction as well to cover the other side.
Example: Find the volume of the solid generated when the region enclosed by and the x -axis is revolved about the y -axis.
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€ y = x
€
, x = 1 , x = 4 ,
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Example: Find the volume of the solid generated when the region enclosed by and x = 2 is revolved about the y -axis.
€ € y = 2 x , y = − 2 x + 4 ,
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Example: Find the volume of the solid generated when the region enclosed by revolved about the x -axis.
€ y = 2 x and y = x
2 is
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Example: Find the volume of the solid generated when the region enclosed by x = 1 is revolved about the line
€ x = − 1 .
€ € y = x
3 , y = 0 , and
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Example: Find the volume of the solid generated when the region enclosed by revolved about the line y = 3 .
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€ x = y
2 and
€ y = x − 2 is
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