Volume of a Hollow Cylinder vs Cylindrical Shell

[CStudy]

In my physics lab, I am asked to calculate the volume of a hollow cylinder. The equation for the volume hollow cylinder below was given. Then, my curiosity made me wonder, is the volume of the hollow cylinder the same as the volume of a cylindrical shell used in calculus? At first though you would assume the answer as yes. However, I have tested this theory using various measurements, resulting in two different results. Can anyone help me understand why Hallow Cylinder does not equal Cylindrical Shell? or maybe disprove my results.

Hollow Cylinder =
(π)(height)((r_o)2−(r_i)2)
Cylindrical Shell = 2(π)(r_i)(height)(thickness)

The subscript "o" means outer-radius, and "i" means inter-radius

 

[berkeman]

In my physics lab, I am asked to calculate the volume of a hollow cylinder. The equation for the volume hollow cylinder below was given. Then, my curiosity made me wonder, is the volume of the hollow cylinder the same as the volume of a cylindrical shell used in calculus? At first though you would assume the answer as yes. However, I have tested this theory using various measurements, resulting in two different results. Can anyone help me understand why Hallow Cylinder does not equal Cylindrical Shell? or maybe disprove my results.

Hollow Cylinder = (π)(height)((r_o)^2−(r_i)^2)

Cylindrical Shell = 2(π)(r_i)(height)(thickness)

The subscript "o" means outer-radius, and "i" means inter-radius

Welcome to the PF.

In the quoted text, I've fixed the r^2 terms.

The first equation is correct for the volume of a hollow cylinder. The second equation is used in calculus to calculate volumes, but what is the key assumption when it is used? You cannot use it for a cylindrical shell of a finite thickness...

 

[CStudy]

The assumption is the cylindrical shell's thickness is infinitesimally small. I guess if you think about it, if you was to cut a hollow cylinder down the middle the surface are of one side would not equal the surface area of the other, unless the thickness was extremely, extremely, extremely small. thanks

 

[SteamKing]

In my physics lab, I am asked to calculate the volume of a hollow cylinder. The equation for the volume hollow cylinder below was given. Then, my curiosity made me wonder, is the volume of the hollow cylinder the same as the volume of a cylindrical shell used in calculus? At first though you would assume the answer as yes. However, I have tested this theory using various measurements, resulting in two different results. Can anyone help me understand why Hallow Cylinder does not equal Cylindrical Shell? or maybe disprove my results.

Hollow Cylinder =
(π)(height)((r_o)2−(r_i)2)
Cylindrical Shell = 2(π)(r_i)(height)(thickness)

The subscript "o" means outer-radius, and "i" means inter-radius

Well, without access to your results, I can't say if you've done your calculations correctly.

However, the volume of the cylindrical shell, V_shell = 2πrht, is accurate enough when t << r. This volume is calculated knowing the circumference of the cylinder, which is 2πr, and then multiplying that by the height to get the surface area, 2πrh,and then multiplying the surface area by the thickness t to get the volume.

Let's take a case where h = 1 and r_o = 1, and let r_i vary a bit:

ro   ri     Vcyl   Vshell  % Diff.
1    0.90  0.5969  0.6283  5.26
1    0.95  0.3063  0.3142  2.57
1    0.99  0.0625  0.0628  0.50

As you can see here, the closer r_i comes to r_o, the closer the volume of the shell comes to the volume of the hollow cylinder.

 

[CStudy]

Awesome explanation.