Is x^2 + y^2 = 4R^2 a Cylinder or a Circle in 3D Space?

  • Thread starter whatisreality
  • Start date
In summary, the conversation discusses the volume V inside a cylinder described by the equation x^2 + y^2 = 4R^2 and between z = (x^2 + 3y^2)/R and the (x,y) plane. The equation is for a circle in 3D space, creating an infinite cylinder when stacked. The value of z is arbitrary in this equation. The conversation also clarifies the concept of arbitrary values and their significance in equations.
  • #1
whatisreality
290
1
I don't think this goes in the homework section because I don't actually want help answering the question, I want to know what it means!

Consider the volume V inside the cylinder x2 +y2 = 4R2 and between z = (x2 + 3y2)/R and the (x,y) plane, where x, y, z are Cartesian coordinates and R is a constant. Write down a triple integral for the volume V using cylindrical coordinates. Include the limits of integration (three upper and three lower). Evaluate the integral to determine the volume V in terms of R.

My main problem is when it asks about the cylinder x2 +y2 = 4R2. I'm nearly 100% sure that equation is not actually for a cylinder but for a circle! And I'm not entirely clear on whether I'm integrating two shapes, as in two volume integrals, or it's describing just one big shape.

In the latter case, I still don't know where cylinders come into it.
 
Mathematics news on Phys.org
  • #2
The equation is for a circle but you also have that ##z## varies between the ##xy##-plane and ##(x^2+3y^2)/R##. If you stack a lot of circles on top of each other you get a cylinder. So the first equation only describe one part of the cylinder while the third coordinate, ##z## is free too change value.
So the equation is a circle if you were in a plane, If you were in 3d-space you have an infinite cylinder if you didn't have any restrictions on ##z##.
 
  • Like
Likes whatisreality
  • #3
I draw the shape of that cylinder with [itex] R=1 [/itex].
 

Attachments

  • 3dplot-page-001.jpg
    3dplot-page-001.jpg
    42.9 KB · Views: 338
  • Like
Likes whatisreality
  • #4
whatisreality said:
Consider the volume V inside the cylinder x2 +y2 = 4R2 and between z = (x2 + 3y2)/R and the (x,y) plane, where x, y, z are Cartesian coordinates and R is a constant.

Incand said:
The equation is for a circle but you also have that ##z## varies between the ##xy##-plane and ##(x^2+3y^2)/R##. If you stack a lot of circles on top of each other you get a cylinder. So the first equation only describe one part of the cylinder while the third coordinate, ##z## is free too change value.
So the equation is a circle if you were in a plane, If you were in 3d-space you have an infinite cylinder if you didn't have any restrictions on ##z##.
In three dimensions (which is implied by the statement that x, y, and z are coordinates), the equation (##x^2 + y^2 = 4R^2##) is a right circular cylinder. Since z does not appear in the equation, it is arbitrary.
 
  • #5
Mark44 said:
In three dimensions (which is implied by the statement that x, y, and z are coordinates), the equation (##x^2 + y^2 = 4R^2##) is a right circular cylinder. Since z does not appear in the equation, it is arbitrary.
Arbitrary as opposed to zero?
 
  • #6
whatisreality said:
Arbitrary as opposed to zero?
"Arbitrary" means "any value."
 
  • #7
Mark44 said:
"Arbitrary" means "any value."
I know. So arbitrary means it can take the value zero and others, as opposed to just zero, which is what I thought the equation meant. Shouldn't the z appear somewhere in the equation though? I feel like this is quite a basic concept I've misunderstood or missed! Oops!
 
  • #8
whatisreality said:
I know. So arbitrary means it can take the value zero and others, as opposed to just zero, which is what I thought the equation meant. Shouldn't the z appear somewhere in the equation though? I feel like this is quite a basic concept I've misunderstood or missed! Oops!
No, z doesn't have to appear in the equation. In the plane, the equation x = 2 is a vertical line. Here, y is not mentioned, and it is arbitrary, so every point in the plane with coordinates (2, y) is a point on this line. The situation is similar for your cylinder equation.
 
  • Like
Likes PeroK and whatisreality
  • #9
Oh, I get it! Ok, that is a really important thing to know. Also know what the question is asking now! Thanks for your help :)
 

Related to Is x^2 + y^2 = 4R^2 a Cylinder or a Circle in 3D Space?

1. What does it mean when someone says "I can't interpret this question"?

When someone says "I can't interpret this question," it means that they are having difficulty understanding or making sense of the question being asked. They may not understand the language used, the context of the question, or the question may be too vague or complex for them to interpret.

2. Why would someone not be able to interpret a question?

There are several reasons why someone may not be able to interpret a question. It could be due to a language barrier, lack of knowledge or understanding of the subject matter, or the question may be poorly worded or confusing. Additionally, individuals with learning or cognitive disabilities may struggle with interpreting questions.

3. How can someone improve their ability to interpret questions?

Improving one's ability to interpret questions can be done by practicing active listening and reading comprehension skills. It also helps to ask for clarification if a question is unclear or to break down complex questions into smaller, more manageable parts. Additionally, learning new vocabulary and understanding the context of a question can aid in interpretation.

4. What should I do if I can't interpret a question on a test or exam?

If you encounter a question on a test or exam that you cannot interpret, it is important to remain calm and not panic. First, try to read the question again and break it down into smaller parts to see if you can understand it better. If you are still struggling, do not hesitate to ask the teacher or professor for clarification. They are there to help you and want you to succeed.

5. Can someone's interpretation of a question be different from someone else's?

Yes, it is possible for someone's interpretation of a question to be different from someone else's. This can be due to differences in language proficiency, cultural context, or individual perspectives and experiences. It is important to be open-minded and understand that there can be multiple valid interpretations of a question.

Similar threads

Calculate this Integral around the Circular Path using Green's Theorem
    • Calculus and Beyond Homework Help
Replies
3
Views
560
B Invariant under rotation: Banal, obvious, or noteworthy?
    • General Math
Replies
2
Views
1K
MHB Nour's question via email about volume of revolution using cylindrical shells.
    • General Math
Replies
1
Views
4K
I Atmospheric Density vs Altitude in an O'Neil Cylinder
    • Classical Physics
Replies
1
Views
791
Volume of the smaller part of a sliced cylinder
    • Calculus and Beyond Homework Help
Replies
10
Views
2K
Surface integral: Calculate the heat flow from a cylinder
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
MHB Triple Integrals: Finding Volume of Solid S Bounded by Planes
    • Calculus
Replies
5
Views
2K
MHB Brenton's questions via email about volume by revolution
    • General Math
Replies
1
Views
4K
MHB Edin's question via email about volume by revolution.
    • General Math
Replies
0
Views
5K
Finding the Centre of Mass of a Hemisphere
    • Calculus and Beyond Homework Help
Replies
9
Views
1K

Hot Threads

Recent Insights

Back
Top