Torricelli's Trumpet Proof using Method of Indivisibles

In summary, the conversation is about someone struggling with understanding Torricelli's trumpet without using integrals. They are looking for help and resources to better understand the concept. A historical notes document is provided as a reference.
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BlackTiger
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Hey am currently struggling with the proof for Torricelli's trumpet (finite volume with infinite surface area for function y=1/x about the x-axis for x > or = to 1) without using the modern notation, i.e., integrals. Any help would be appreciated...

i have found a couple of explanations for using the method of indivisibles but they seem to be few and far between with a lot of it being very very basic and not really giving an understanding of how torricelli himself worked it out before then proving it using integrals in the modern day way so i am looking for any help, be it website links or your own knowledge to point me in the right direction for figuring this all out

many thanks
 
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1. What is Torricelli's Trumpet Proof using Method of Indivisibles?

Torricelli's Trumpet Proof is a mathematical concept developed by Italian physicist Evangelista Torricelli in the 17th century. It uses the Method of Indivisibles, a precursor to calculus, to prove that the volume of a trumpet-shaped object with infinite length and a finite opening is equal to the volume of a sphere with the same radius as the opening.

2. How does the Method of Indivisibles work?

The Method of Indivisibles works by dividing a continuous shape into an infinite number of infinitesimally small parts, or "indivisibles". By summing the areas or volumes of these indivisibles, the total area or volume of the original shape can be calculated.

3. What is the significance of Torricelli's Trumpet Proof?

Torricelli's Trumpet Proof is significant because it demonstrates the power and usefulness of the Method of Indivisibles in solving complex mathematical problems. It also provides a theoretical basis for the concept of infinity and helps to bridge the gap between geometry and calculus.

4. Are there any real-life applications of Torricelli's Trumpet Proof?

While Torricelli's Trumpet itself is a theoretical concept, the Method of Indivisibles has been applied in various fields of science and mathematics, such as fluid dynamics and probability theory. It has also been used to solve real-world problems, such as determining the volume of irregularly shaped objects.

5. Are there any limitations to Torricelli's Trumpet Proof?

One limitation of Torricelli's Trumpet Proof is that it only works for objects with infinite length and a finite opening. It does not apply to objects with finite length or infinite openings. Additionally, the Method of Indivisibles is not a rigorous mathematical proof and has been criticized for its use of infinitesimals, which were not fully understood at the time of Torricelli's work.

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