Problem Deriving Volume of a Cylindrical shell

In summary, the individual was trying to derive a formula for the volume of a cylindrical shell of inner radius r and outer radius dr+r and length h. However, they could not get the formula in the form h2(pi)rdr starting from where they did. Instead, they ended up getting r^2 + 2r*dr + dr^2, which is pretty close to the correct formula.
  • #1
My4rk89
6
0
I have seen the derivation for the formula:

h2(pi)rdr used in math textbooks. However, earlier today I had a physics problem where I needed to use the volume of a cylindrical shell of inner radius r and outer radius dr+r and length h. without remembering the formula I tried to derive it starting by determining the area of the base minus the area of the gap:
A=(pi)r^2-(pi)(r+dr)^2

and then multiplying by h


I could not get this in the form h2(pi)rdr starting from where I did.

I understand the derivation given here:
http://www.stewartcalculus.com/data...texts/upfiles/3c3-Volums-CylinShells_Stu .pdf

But I'm just frustrated that I can't understand why the method I tried doesn't reduce to the correct formula. Its simply the area of the base*height!

Why can't that formula be arrived at by my method?
 
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  • #2
My4rk89 said:
I have seen the derivation for the formula:

h2(pi)rdr used in math textbooks. However, earlier today I had a physics problem where I needed to use the volume of a cylindrical shell of inner radius r and outer radius dr+r and length h. without remembering the formula I tried to derive it starting by determining the area of the base minus the area of the gap:
A=(pi)r^2-(pi)(r+dr)^2

and then multiplying by h
Assuming that dr is positive, what you'll get from your formula is a negative number.

So instead, you should have A = pi(r + dr)^2 - pi*r^2 = pi*( (r + dr)^2 - r^2).

Expand the (r + dr)^2 part and combine like terms. What do you get?
My4rk89 said:
I could not get this in the form h2(pi)rdr starting from where I did.

I understand the derivation given here:
http://www.stewartcalculus.com/data...texts/upfiles/3c3-Volums-CylinShells_Stu .pdf

But I'm just frustrated that I can't understand why the method I tried doesn't reduce to the correct formula. Its simply the area of the base*height!

Why can't that formula be arrived at by my method?
 
  • #3
When I expand I get r^2 +2rdr + dr^2

so the r^2 drops out. but what happens to the the dr^2 ?
 
  • #4
The idea here is that if dr is small in comparison to r, the (dr)^2 will be very much smaller and can be discarded.

For example, if r = 1 and dr = .01, then (r + dr)^2 = (1.01)^2 = 1.0201.

If I expand (r + dr)^2, I get r^2 + 2r*dr + (dr)^2 = 1 + 2(.01) + (.0001). If I omit the (dr)^2 term, I get 1.02, which is pretty close to 1.0201.

The smaller dr is in comparison to r, the closer (r + dr)^2 is to r^2 + 2r*dr.
 
  • #5
Alright that's what I thought I just couldn't find any verification!

I never really encountered a differential element squared before.

Thanks so much!
 

1. What is a cylindrical shell?

A cylindrical shell is a three-dimensional geometric shape that is formed by the lateral surface of a cylinder. It has two circular bases and a curved surface that connects the two bases.

2. What is the formula for finding the volume of a cylindrical shell?

The formula for finding the volume of a cylindrical shell is V = πr2h, where r is the radius of the base and h is the height of the cylinder.

3. How do you derive the formula for the volume of a cylindrical shell?

The volume of a cylindrical shell can be derived by taking the difference between the volumes of two cylinders. The first cylinder has a radius r and height h, while the second cylinder has a radius r and height (h - x). The difference in volumes is then multiplied by π to get the final formula.

4. Can the volume of a cylindrical shell be negative?

No, the volume of a cylindrical shell cannot be negative. It is a physical quantity and cannot have a negative value. If the result of the formula is negative, it means that the height is greater than the radius, which is not possible for a cylindrical shell.

5. How is the volume of a cylindrical shell used in real life?

The volume of a cylindrical shell is used in many real-life applications, such as calculating the volume of cylindrical containers like soda cans or water bottles. It is also used in engineering and construction to determine the amount of material needed for cylindrical structures, such as pipes or columns.

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