Need help, can someone explain cylindrical shell method for volume?

In summary, the cylindrical shell method is a technique used in calculus to find the volume of a solid of revolution by dividing the shape into thin cylindrical shells and summing their volumes. To set up the integral for this method, the height and circumference of each shell must be determined and then integrated over the desired bounds. An example of using this method is finding the volume of a solid created by rotating the region between two curves. Advantages of the cylindrical shell method include its ability to find the volume of irregularly shaped solids, but it can be more difficult to visualize and set up integrals for compared to other methods. Other applications of this method include finding the moment of inertia and surface area, but it is primarily used for finding volume in calculus
  • #1
timm3r
10
0
im stuck and my book doesn't explain much on how to get the volume of a shape by the cylindrical shell method. What i really need is to show how to get the general formula of a volume of a sphere and a cone using cylindrical shells.
 
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  • #2
http://pages.pomona.edu/~sg064747/teaching/F06-31/Lecture/F06-31-Lecture08.pdf
 
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  • #3
wow thanks for the quick response,
 
  • #4
I would also like to have it explain to me but the website given is out of order.
 

1. What is the cylindrical shell method for finding volume?

The cylindrical shell method is a technique used in calculus to find the volume of a solid of revolution, where a 2D shape is rotated around an axis to create a 3D shape. It involves dividing the shape into thin cylindrical shells and summing their volumes to approximate the total volume of the solid.

2. How do you set up the integral for the cylindrical shell method?

To set up the integral for the cylindrical shell method, you first need to determine the height of each cylindrical shell. This is typically the distance between the axis of rotation and the function being rotated. Then, you need to find the circumference of each shell by multiplying the radius (distance from the axis of rotation) by 2π. Finally, you integrate the product of the height and circumference over the desired bounds to get the total volume.

3. Can you provide an example of using the cylindrical shell method to find volume?

Sure! Let's say we want to find the volume of a solid created by rotating the region between the curves y = x^2 and y = 0 from x = 0 to x = 1. First, we would set up the integral as ∫(2πx)(x^2)dx, where 2πx represents the circumference and x^2 represents the height of each cylindrical shell. Then, we integrate from x = 0 to x = 1 to get the volume of the solid.

4. What are the advantages and disadvantages of using the cylindrical shell method?

One advantage of using the cylindrical shell method is that it can be used to find the volume of irregularly shaped solids, whereas other methods such as the disk or washer method require the shape to have a specific symmetry. However, it can be more difficult to visualize and set up the necessary integrals for the cylindrical shell method compared to other volume-finding methods.

5. Are there any other applications of the cylindrical shell method?

Yes, the cylindrical shell method can also be used in other areas such as physics and engineering to find the moment of inertia of an object, which is a measure of its resistance to rotational motion. It can also be used to find the surface area of a solid of revolution by summing the surface areas of each cylindrical shell. However, the method is primarily used in calculus for finding volume.

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