Confused about using the integral and infinitesimal?

In summary, the conversation discusses finding the volume of a sphere using infinitesimals instead of the standard formula for integration. The concept involves stacking infinitesimally thin disks to approximate the volume and finding the thickness of spherical shells. This method simplifies the formula for one "slice" and allows for integration to find the volume of the sphere.
  • #1
Dufoe
4
0
I don't understand what is meant by "derive the formula for finding the volume of a sphere that uses infinitesimals but not the standard formula for the integral"?
Is this talking about Gauss or what? I'm completely self taught in calculus and I did three proofs already... the old cylinder / cone proof, and the other two used the standard formula for the integral.. Even any link to something that explains this would help. I'm only aware of 7 proofs and I don't get how any of them meet both criteria..? Thanks!
 
Physics news on Phys.org
  • #2
Possibly the exact wording of the question would help.
 
  • #3
This may help:

http://math.stackexchange.com/quest...phere-by-dividing-it-into-infinitesimal-cubes

The idea is you already know the area of a cross-section of a sphere (a disk) from geometry: πr2. So now, what happens if you try to stack disks on top of each other, making their height increasingly small (or, conversely, having a greater and greater number of thinner disks).
 
  • #4
Yup, as Hector is alluding to, you need to find the thickness of spherical shells. If you find the area of a shell and then multiply by a infinitessimal thickness you get the volume of the infinitessimally thick shell. Integrate these shell volumes to find the volume of the sphere. The idea to realize is that since the thickness of the shell is infinitessimal the shell area is the same in the inner and outer surface of the shell. as the thickness of the shell decreases the inner surface and outer surface areas get closer together. So for an infinitessimal thickness they are the same area.
 
  • #5
I would start by observing that the volume of a sphere is twice the volume of half a sphere. This makes the formula for one "slice" much simpler.
 

Related to Confused about using the integral and infinitesimal?

1. What is the difference between an integral and an infinitesimal?

An integral is a mathematical tool used to find the area under a curve or the volume of a solid. It is represented by the symbol ∫ and is used to sum up an infinite number of infinitesimal values. Infinitesimals are infinitely small quantities that are used in calculus to represent the change in a function over a very small interval.

2. How do I know when to use an integral and when to use an infinitesimal?

An integral is used when you need to find the overall change in a function over a certain interval. This can be represented by the area under a curve on a graph. Infinitesimals, on the other hand, are used when you need to find the instantaneous rate of change at a specific point on a function. This can be represented by the slope of a tangent line on a graph.

3. Can an integral and an infinitesimal be used interchangeably?

No, an integral and an infinitesimal cannot be used interchangeably. They serve different purposes in mathematics and have different symbols and notations. While both are used in calculus, they are not interchangeable and should be used appropriately depending on the problem at hand.

4. Are there any special rules for using integrals and infinitesimals?

Yes, there are certain rules and techniques for using integrals and infinitesimals in calculus. These include the power rule, the chain rule, and integration by parts. It is important to familiarize yourself with these rules and practice using them in order to properly solve calculus problems involving integrals and infinitesimals.

5. How can I improve my understanding of using integrals and infinitesimals?

The best way to improve your understanding of using integrals and infinitesimals is to practice solving problems and to seek help from a teacher or tutor if you are struggling. You can also use online resources, such as videos and interactive tutorials, to supplement your learning. It is important to have a solid understanding of the fundamentals of calculus before attempting to use integrals and infinitesimals in more complex problems.

Similar threads

B How can hyperreal numbers make infinitesimals logically sound in calculus?
    • Calculus
Replies
24
Views
3K
B Question about volume of a sphere
    • Calculus
Replies
16
Views
581
B Using infinitesimals to find the volume of a sphere/surface
    • Calculus
Replies
8
Views
2K
B Why is the definite integral of 1/x from -1 to 1 undefined?
    • Calculus
Replies
4
Views
3K
B Line Integral, Dot Product Confusion
    • Calculus
Replies
8
Views
2K
I How to derive this log related integration formula?
    • Calculus
Replies
2
Views
1K
I Deriving the volume of a sphere using semi-circles
    • Calculus
Replies
5
Views
2K
What is the purpose of a surface integral and how is it calculated?
    • Calculus
Replies
6
Views
2K
B Nature of Limits: Learn Calculus Basics w/o Limits Distraction
    • Calculus
Replies
13
Views
2K
Integration by Parts confusions
    • Calculus
Replies
9
Views
2K

Hot Threads

Recent Insights

Back
Top