Surface area of a revolution, why is this wrong?

In summary, the conversation discusses the concept of using Δx as an approximation for finding the area under a curve. However, it is explained that this method is incorrect as it does not take into consideration the shape of the curve. The example of a surface of revolution is used to show that Δx is not a reliable approximation for the distance between points on a curve.
  • #1
CollinsArg
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Why is this way of thinking wrong?. can't I assume that when Δx tends to zero is a sufficient approximation of what I want to get? It confuses me with the basic idea of integrating a function to get the area beneath a curve of a function (which isn't also as perfect) .

PD: I put Δx tends to infinity in the image but I think the right way I was thinking is to zero.

Thank you. ç

(English is not my first language).
 

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  • #2
This may help your understanding a bit



Notice they use the Pythagorean theorem to get the length of the dxdy triangles hypotenuse with ##dx^2 +dy^2##which gets transformed into ##1+f’^2## When you factor out the dx and then of course you take its square root and integrate it from x0 to x2.
 
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  • #3
CollinsArg said:
Why is this way of thinking wrong?. can't I assume that when Δx tends to zero is a sufficient approximation of what I want to get?
Yes, your way of thinking is wrong. You have to take into account the shape of the curve. Your way would work if you were dealing with a cylinder, but it doesn't work for a surface of revolution. ##\Delta x## is not a good approximation of the distance between the points ##(x, f(x))## and ##(x + \Delta x, f(x + \Delta x)##. Take a look at the video that @jedishrfu showed.
 

Related to Surface area of a revolution, why is this wrong?

1. What is the surface area of a revolution?

The surface area of a revolution is the area created when a 2-dimensional shape is rotated around an axis to form a 3-dimensional shape. It is commonly used in calculus to find the surface area of curved objects.

2. Why is the surface area of a revolution calculation sometimes considered wrong?

The surface area of a revolution calculation can be considered wrong because it is an approximation rather than an exact measurement. It relies on dividing the curved surface into infinitely small sections, which can result in errors and inaccuracies.

3. What are some common mistakes made when calculating the surface area of a revolution?

Some common mistakes include not using the correct formula for the specific shape being rotated, not considering the thickness of the shape, and not using small enough sections to accurately approximate the curved surface.

4. How can the accuracy of the surface area of a revolution calculation be improved?

The accuracy of the calculation can be improved by using smaller sections and a larger number of sections, as well as using the correct formula for the specific shape being rotated. Additionally, using advanced mathematical techniques such as integral calculus can provide more precise results.

5. In what real-world applications is the surface area of a revolution important?

The surface area of a revolution is important in various fields such as engineering, architecture, and physics. It is used to calculate the surface area of objects like pipes, containers, and curved structures, which is essential for designing and constructing these objects with the correct dimensions and materials.

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