Volume Calculation Using Cylindrical Shells

In summary, the conversation discusses the use of the method of cylindrical shells to find the volume of a solid obtained by rotating a region bounded by two given curves about the x-axis. The problem is set up using an integral and the height and radius of the "shell" are determined. There is a question about the notation "(y-2)**2" and it is clarified that it should be written as (y-2)^2. The problem is eventually solved.
  • #1
quickclick330
83
0
Consider the given curves to do the following.
x = 3 + (y-2)**2, x = 4
Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis.


V = ??

****************

I set up the problem like this...

V = integral from 0 to 3((4-(y^2-4y+7))(y + 1) dy)


(height) (radius) ...of "shell"

Did I set this up right? Hopefully this is clear enough, if you can't understand I can clarify. Thanks for the help in advance!
 
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  • #2
What does "(y-2)**2" mean? Did you mean [tex](y-2)^2[/tex]?
 
  • #3
yes, sorry, it copy-pasted like that
 
  • #4
never mind, i solved it finally
 

1. What is the formula for finding the volume of a cylinder using cylindrical shells?

The formula for finding the volume of a cylinder using cylindrical shells is V = 2πrh, where V is the volume, r is the radius, and h is the height of the cylinder.

2. How is the shell method different from the disk method?

The shell method involves slicing the solid into infinitesimally thin cylindrical shells and summing their volumes, while the disk method involves slicing the solid into infinitesimally thin disks and summing their volumes.

3. When should I use the cylindrical shell method to calculate volume?

You should use the cylindrical shell method when the shape being rotated around the axis of rotation is a function of the height of the cylinder and the axis of rotation is perpendicular to the base of the cylinder.

4. Can the cylindrical shell method be used to find the volume of any solid?

No, the cylindrical shell method can only be used to find the volume of solids that can be generated by rotating a function around an axis of rotation that is perpendicular to the base of the solid.

5. How do I set up the integral for finding the volume using cylindrical shells?

To set up the integral for finding the volume using cylindrical shells, you need to integrate the formula V = 2πrh over the desired interval, with r and h expressed in terms of the variable of integration. This will give you the total volume of the solid.

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