Integration by filaments or integration by strate?

In summary, the problem is broken into two integrals, one from ##x=-2## to ##x=-\sqrt{2}## and one from ##x=-\sqrt{2}## to ##x=-1##. The volume of the infinitesimal slice is given by the sum of the volumes of the two disks.
  • #1
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Homework Statement
look at the image
Relevant Equations
cylindrical coordinates
integration by strates
integration by filaments
Greetings
While solving the following exercice, ( the method used is the integration by filaments and I have no problem doing it this way)
1622996959713.png

here is the solution
1622997058481.png

1622997162664.png

My question is the following:
I want to do the integration by strate and here is my proposition
1622997349308.png

is that even correct?
I would like to know if there is cases when switching between the two methods of integration is not possible?
thank you in advance!
 
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  • #2
I've never heard of "integration by strate." Could you explain what this means?
 
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  • #3
Certainly look at the image
1623001742727.png
it means doing the integration by rings and then integrating by hight
 
  • #4
Oh, okay. In the US, I think that technique is referred to as the disc method.
 
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  • #5
You should be able to use that method, but you're going to have to break the integral up into two integrals, one from ##x=-2## to ##x=-\sqrt{2}## and one from ##x=-\sqrt{2}## to ##x=-1##.
 
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  • #6
thank you very much , indeed I used your insights to solve the question under the disk method
1623071002384.png

but the only way I could find out was to
1623071123825.png


and then substructing the sphere from the paraboloid
I would be very grateful if you could elaborate more about the setting of your method and why you need to split the integration this way?
thank you!
 
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  • #7
To keep it simple, I'm going to consider volume integrals but the same ideas apply to your problem.

When you do the integrals over ##r## and ##\theta##, you end up with an integral of the form
$$\int_{x_1}^{x_2} \pi r^2 \,dx$$ where we can identify ##\pi r^2## as the area of the disk and ##dV = \pi r^2\,dx## as the volume of the infinitesimal slice. In this particular problem, from ##x=-2## to ##x=-\sqrt{2}##, the volume is given by an integral of that form. However, from ##x=-\sqrt{2}## to ##x=-1##, you have a disk with a hole in the middle with area ##\pi(r_{\rm outer}^2-r_{\rm inner}^2)##. Hence, the volume of that portion is given by
$$\int_{x_1}^{x_2} \pi (r_{\rm outer}^2-r_{\rm inner}^2) \,dx.$$ Because you have two different integrands, you need to do the integrals for each range separately.

Note that you can manipulate your integrals similarly:
\begin{align*}
I &= \int_{-2}^{-1} \int_0^{2\pi} \int_0^{\sqrt{2-x^2}} xr\,dr\,d\theta\,dx
-\int_{-\sqrt 2}^{-1} \int_0^{2\pi} \int_0^{\sqrt{x+2}} xr\,dr\,d\theta\,dx \\
&= \int_{-2}^{-\sqrt{2}} \int_0^{2\pi} \int_0^{\sqrt{2-x^2}} xr\,dr\,d\theta\,dx
+\int_{-\sqrt{2}}^{-1} \int_0^{2\pi} \int_0^{\sqrt{2-x^2}} xr\,dr\,d\theta\,dx
-\int_{-\sqrt 2}^{-1} \int_0^{2\pi} \int_0^{\sqrt{x+2}} xr\,dr\,d\theta\,dx \\
&= \int_{-2}^{-\sqrt{2}} \int_0^{2\pi} \int_0^{\sqrt{2-x^2}} xr\,dr\,d\theta\,dx
+\int_{-\sqrt{2}}^{-1} \int_0^{2\pi} \int_{\sqrt{x+2}}^{\sqrt{2-x^2}} xr\,dr\,d\theta\,dx
\end{align*} The limits on the ##r## integral are different for ##x=-2## to ##x=-\sqrt{2}## and for ##x=-\sqrt{2}## to ##x=-1##.
 
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  • #8
thanks a million!
 

1. What is integration by filaments?

Integration by filaments is a computational method used in scientific research to study the behavior of filaments, which are long, thin structures such as proteins or DNA strands. This method involves tracking the position and movement of individual filaments over time to understand their interactions and properties.

2. How does integration by filaments differ from integration by strate?

Integration by filaments focuses on the behavior of individual filaments, while integration by strate looks at the behavior of larger structures or networks made up of multiple filaments. Integration by filaments is more detailed and time-intensive, while integration by strate provides a broader overview of the system.

3. What are the advantages of using integration by filaments?

Integration by filaments allows for a more detailed understanding of the behavior of individual filaments, which can provide insights into larger systems. It also allows for the study of dynamic processes, such as the movement and interactions of filaments over time.

4. How is integration by filaments used in scientific research?

Integration by filaments is used in various fields of science, including biology, physics, and materials science. It is often used to study the structure and function of biological systems, such as the cytoskeleton, as well as the behavior of materials, such as polymers and nanotubes.

5. Are there any limitations to integration by filaments?

Integration by filaments can be limited by the complexity of the system being studied and the amount of data that needs to be processed. It also requires specialized software and expertise, which may not be readily available to all researchers. Additionally, this method may not accurately represent the behavior of filaments in a real-world setting.

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