Method of cylindrical shells

In summary, the torus created by rotating the figure described is off by r^3 because the area of the torus is not taken into account when integrating along the path of revolution.
  • #1
ktpr2
192
0
I know one can figure the volume of a torus by the difference of two volumes, but I'm trying out the method of cylinderical shells. As far as i understand, you can often create a primitive with a calcuable volume and approximate the volume of the shape you wish by scaling the primitive along the curve it creates, adding infinitely many times.

The problem is that my answer is wrong when i try to set up an integral when thinking in terms of cylinderical shells:

We have a rectangle, bent in the shape of a circle, with length [tex]2 \pi r[/tex] height [tex]\sqrt{1-x^2}[/tex] and width[tex] \Delta x[/tex], so it's volume should be all that multiplied together.

I have torus radius (this torus is just a circle, really) of R and the circle being revolved has a radius of r. So my integral is:

[tex]\int_{R-r}^{R+r} 4 \pi x \sqrt{1-x^2} \Delta x[/tex]

and the above is off by [tex]r^3 R[/tex] when i use differences of two volumes, what conceptual flaw am I making?
 
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  • #2
You should better define your intended torus, be more specific with which radius is which, and what you mean by rectangle, the length, and the height with respect to.

http://whistleralley.com/torus/torus.htm does a good explanation of the general integration of a torus.

If you wanted to try again, an easier way to explain your situation (even for yourself) is to refer everything to the coordinate plane, i.e.

A line drawn from the origin along the x-axis to a point (R,0) is the center of a circle with radius r. The volume of the torus created by rotating the figure described is given by integrating the area of the torus along the path of revolution... etc etc
 
  • #3
Hmm, i'll see if casting things in that light simpifies things.
 

What is the Method of Cylindrical Shells?

The Method of Cylindrical Shells, also known as the Shell Method, is a mathematical technique used to calculate the volume of a solid of revolution. It involves breaking down the solid into a series of thin cylinders and then integrating their volumes to find the total volume.

When is the Method of Cylindrical Shells used?

This method is typically used when finding the volume of a solid that is formed by rotating a region bounded by a curve around a vertical or horizontal axis. It is commonly used in calculus and physics to solve problems related to areas and volumes.

What are the advantages of using the Method of Cylindrical Shells?

One of the main advantages of this method is its simplicity compared to other methods. It also allows for the calculation of volumes for solids with irregular shapes or those that cannot be easily solved using other techniques such as the disk method.

What are the limitations of the Method of Cylindrical Shells?

The main limitation of this method is that it can only be used for solids of revolution. It also requires the use of calculus and may be more complex for some individuals to understand compared to other methods.

Can the Method of Cylindrical Shells be used for both horizontal and vertical axis?

Yes, the method can be used for both horizontal and vertical axis as long as the solid is formed by rotating a region around a single axis. The choice of axis will depend on the given problem and the orientation of the solid.

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