Areas of surfaces of revolution

In summary, the conversation discusses the confusion surrounding the use of cylindrical slices to approximate surface areas in integral calculus. The formula for finding surface area using frustum surface area is given, but the question arises as to why cylindrical slices are not used instead. A similar question is found in a random physics forum thread, but no clear answer is given.
  • #1
matts0
11
0
Hello. I just started to study integral calculus not long ago and I have some confusion when it comes to calculating the areas of surfaces of revolution using integral.
As from my testbook, when we want to calculate this kind of surface area, we often use the frustum surface area to approximate it in ds slant. Then integrate it along the defining curve( f(x) ) as the following formula states:
S=∫2πf(x)ds=∫2πf(x)√(1+f'(x)2) dx
However my problem is why we don't use the cylinder slice with dx thick to approximate it and then integrate over the specified region, as we do when calculating the volumes of solids of revolution.
And I found in this case the formula will just be:
S'=∫2πf(x) dx which is different from above.
So please someone tell me where I made a mistake. I am sorry my post is lengthy and I also wish I could explain clearer.
Thanks a lot for your time.
 
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  • #2
However my problem is why we don't use the cylinder slice with dx thick to approximate it and then integrate over the specified region
,

The cylindrical slice of width dx gives the exact volume (not an approximation!) of a cylindrical shell. For a sphere, finding a volume first with cylindrical shells then taking a derivative of the volume will give the surface area of the sphere. But for other non symetric volumes that might not be so easy to do.

read more at a randomly selected thread on the subject
https://www.physicsforums.com/showthread.php?t=528358

That was pure coincidence
 
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  • #3

Related to Areas of surfaces of revolution

What is a surface of revolution?

A surface of revolution is a three-dimensional shape created by rotating a two-dimensional curve around an axis. The resulting surface has rotational symmetry around the axis.

What are some examples of surfaces of revolution?

Some common examples of surfaces of revolution are spheres, cylinders, cones, and tori (donuts). These shapes can be created by rotating a circle, rectangle, triangle, or other closed curve around a central axis.

How are the areas of surfaces of revolution calculated?

The area of a surface of revolution can be calculated using calculus. It involves finding the surface area of a single slice of the surface, and then integrating this over the entire surface. Alternatively, there are formulas specific to each type of surface of revolution that can be used to calculate the area.

What are some real-world applications of surfaces of revolution?

Surfaces of revolution have many practical applications in engineering and design. For example, they are commonly used in the design of car wheels, propellers, and pipes. They are also used in architecture, such as the dome of the Pantheon in Rome.

How do surfaces of revolution relate to calculus?

Surfaces of revolution are closely tied to calculus, as they can be used to demonstrate concepts such as integration and rotational symmetry. Calculus is also used to calculate the area and volume of surfaces of revolution, making it an essential tool for understanding and working with these shapes.

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