Difficulty visualizing given set

  • #1
Physicsdudee
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2
Homework Statement
Hey,
As you can see in the attachment, I am given a Set that I should integrate a function on. The integrating part should not be a problem, however I am having a hard time figuring out how this set looks like and hence which coordinates to use. I get that the best option would probably be spherical coordinates, but I have to have an idea how this set looks like so I can determine the bounds of integration, that is bounds for φ, Θ and R.
Relevant Equations
Spherical coordinates: x=Rsin(Θ)cos(φ), y=Rsin(Θ)sin(φ), z=Rcos(Θ)
Okay so I know, that if the radius is 0, the z coordinate will run from -1 to +1. If the radius tends to one, the z coordinate will tend to 0.
But I still cannot imagine how this set looks like, help would be appreciated.

Thank you.
set.PNG
 
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  • #2
This is the region between the cone [itex]x^2 + y^2 = (z + 1)^2[/itex] and the sphere [itex]x^2 + y^2 + z^2 = 1[/itex] and bounded by the cylinder [itex]x^2 + y^2 = 1[/itex].

I would suggest cylindrical polars.
 
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  • #3
Thanks for your reply.
If I were to use cylindrical coordinates, then I assume I would let φ range from 0 to 2π, z from -1 to 1. But what about R then? R should be a function of the height regarding integration over the cone and then another function of height regarding integration over the sphere right?
 
  • #5
Physicsdudee said:
Thanks for your reply.
If I were to use cylindrical coordinates, then I assume I would let φ range from 0 to 2π, z from -1 to 1. But what about R then? R should be a function of the height regarding integration over the cone and then another function of height regarding integration over the sphere right?

Easier to have [itex]0 \leq r \leq 1[/itex], and the [itex]z[/itex] limits are then conveniently set out in the question...
 
  • #6
Yes, that seems easier, thanks a lot!
 
  • #7
While it's good to try to reason on your own,
be aware that there are useful tools, as @fresh_42 suggests.

Here's another one

https://www.geogebra.org/3d?lang=en
1655490235679.png

1655490273881.png


I'm sure it can also handle spherical and cylindrical coordinates.
 
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  • #9
robphy said:
While it's good to try to reason on your own,
be aware that there are useful tools, as @fresh_42 suggests.

Here's another one

https://www.geogebra.org/3d?lang=en
View attachment 302958
View attachment 302960

I'm sure it can also handle spherical and cylindrical coordinates.
Yep, I know Geo gebra but I was so eager to actually not use anything because in the exam it’s not going to be any different. But I get your point, making use of all these nice tools is extremely good for visualizing
 
  • #10
@Physicsdudee ,
I see that this is the second Thread you have started, but you have not received a formal welcome to Physics Forums. Let's fix that.

:welcome:
 
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  • #11
SammyS said:
@Physicsdudee ,
I see that this is the second Thread you have started, but you have not received a formal welcome to Physics Forums. Let's fix that.

:welcome:
Haha that’s so kind, I appreciate it:)
 
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