Revolution Volume by Cylinder Shells

In summary, the problem involves finding the volume of a region enclosed by two curves when rotated about the x-axis. The upper curve is y=e^x and the lower curve is y=1/x. The method used is the shells method, where the volume is found by multiplying the surface area of the cylinder by the thickness, dx. The radius of the cylinder is x, not y as initially thought.
  • #1
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Rotate about x-axis the region enclosed by y=e^x, y=1/x, x=1 and x=2. I can do the problem with the rings method but I don't how to even set up the integral to solve by the shells method. Help? Thanks
 
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  • #2
The two curves cross well before x= 1 so you have a region in which y=ex is the upper curve and y= 1/x is the lower. That means that the length of your cylinder is exx- 1/x. The radius is x so you cylinder will have a surface area, for each x, of [itex]2\pi x(e^x- 1/x)[/itex]. Multiply that by the "thickness", dx, to find the differential of volume.
 
  • #3
Thanks, but since the region enclosed by the boundaries given is rotated about the x-axis, then, doesn't it mean that the radius of the cylinder is y? I'm a little confused. ??
 

1. What is the concept behind "Revolution Volume by Cylinder Shells"?

The concept of "Revolution Volume by Cylinder Shells" is a mathematical method used to find the volume of a three-dimensional object that has been created by rotating a two-dimensional shape around a specific axis. This method is based on the use of cylindrical shells, which are infinitely thin cylinders stacked together to form the shape of the object.

2. How is the volume calculated using "Revolution Volume by Cylinder Shells"?

The volume is calculated by dividing the shape into infinitely thin cylindrical shells, finding the volume of each shell, and then adding up all the individual volumes. The formula used is V = ∫2πrh dx, where r is the radius of the shell, h is the height of the shell, and dx is the thickness of the shell.

3. What are the advantages of using "Revolution Volume by Cylinder Shells" over other methods?

One of the main advantages of this method is that it can be used to find the volume of irregular shapes that cannot be easily calculated using other methods such as the disk or washer method. It also allows for more precise calculations as the number of shells used can be increased to get a more accurate result.

4. Are there any limitations to using "Revolution Volume by Cylinder Shells"?

While this method is useful for finding the volume of many shapes, it does have some limitations. It is not suitable for shapes with holes or voids, and it can be more time-consuming and complex to use compared to other methods for simpler shapes.

5. How is "Revolution Volume by Cylinder Shells" applied in real-world situations?

This method has various applications in fields such as engineering, architecture, and physics. It can be used to calculate the volume of objects like bottles, pipes, and even buildings with curved surfaces. It is also used in fluid mechanics to determine the volume of liquids in containers with irregular shapes.

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