# Integration via Foliation

• bolbteppa
In summary, the conversation discusses the concept of integration by foliation and the search for resources to further explore this topic. The conversation also mentions the potential connection to surface area of revolution, volumes by cross-sectional areas, and volumes of revolution. Two resources, "Volumes of Solids of Revolution. A Unified Approach" and articles by Walter Carlip and Eric Key, are suggested for further reading. However, it is noted that these resources may not specifically address the foliation aspect of the question.

#### bolbteppa

I posted a question on stack about generalizing the methods of volumes by slicing & volumes of revolutions & came across the notion of integration by foliation. I'm wondering if there are any resources you guys know of where I could delve into this further, I'd love to think in a sophisticated & unified manner about ideas like surface area of revolution, volumes by cross-sectional areas, volumes of revolution, etc... but don't know what to do or where to go, & I'm sure there's something out there on these topics, thanks

You may take a look at Volumes of Solids of Revolution. A Unified Approach which cites the article by Walter Carlip, Disks and Shells Revisited and Eric Key, Disk Shells and inverse functions, both collected in A calculus collection

There is an area of differential geometry that has the concept of foliations, but these are not exactly the same type of foliations mentioned in the stackexchange answer.

Thanks a lot for the links, although they don't target the foliation aspect of my question they are great in and of themselves.

## 1. What is Integration via Foliation?

Integration via Foliation is a mathematical technique used to solve certain types of differential equations. It involves breaking down a complex system into simpler, one-dimensional systems, and then integrating them to find a solution to the original equation.

## 2. What are the advantages of using Integration via Foliation?

Integration via Foliation can simplify the process of solving complex differential equations, making it easier to find a solution. It can also provide insight into the behavior of the system and help identify patterns or relationships between variables.

## 3. How is Integration via Foliation different from other integration methods?

Integration via Foliation is different from other integration methods in that it focuses on breaking down a system into simpler components and integrating them separately. This is in contrast to other methods such as the Riemann sum or the trapezoidal rule, which involve approximating the integral using geometric shapes.

## 4. What types of problems can be solved using Integration via Foliation?

Integration via Foliation is most commonly used to solve partial differential equations, particularly those that involve multiple independent variables. It is also useful for systems of equations that can be broken down into one-dimensional equations.

## 5. Are there any limitations to Integration via Foliation?

One limitation of Integration via Foliation is that it can only be applied to certain types of differential equations. It may not be suitable for all problems, and other integration methods may be more appropriate. Additionally, Integration via Foliation can be computationally intensive and may not always provide an exact solution.