math question: partial surface area of sphere

SnoPearL69

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Aug 26, 2004
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I need to find the surface area of a portion of a sphere with known radius r. What is the surface area a distance h from the surface? (Like, slicing a ball a distance h from the surface, and finding the surface area of that sliced off portion)
 

Heisenberg

Lifer
Dec 21, 2001
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I still don't really understand the question. Can you rephrase it? I would just integrate over the surface using the appropriate limits for the section you want.
 

simms

Diamond Member
Sep 21, 2001
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Surface area = 4pi*r^2.

When you slice it off, it becomes pi r^2.

Thing is, when you slice off a part, that part isn't a circle, so it's hard to calculate the surface area of that...
 

agnitrate

Diamond Member
Jul 2, 2001
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Originally posted by: SnoPearL69
(Like, slicing a ball a distance h from the surface, and finding the surface area of that sliced off portion)

Wow, can you get any more vague?

A distance h in all directions, a radius of h, or what? If it's a distance h, only integrate over the angle of theta that this distance falls upon. If it's a radius, that's easy. Please elaborate on your homework problem a little further.

 

SnoPearL69

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Aug 26, 2004
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Okay, upon further thought of the problem, this can either be solved in 3D (sphere), or approximated in 2D (circle of radius r).

In 3D, I'm talking about a cut of distance h from the surface, towards the center (along a radius). So, imagine sawing off the skull cap of someone's head, and you're cutting a distance of h from the very top towards the center. I need the surface area of that.

In 2D, it's similar, except I'm looking for the area of a section of the circle, again a distance h from the border towards the center, along a radius. For this, imagine a sunrise, and the portion of the sun that has appeared over the horizon is a length of h from top of the arch to the plane of the horizon.

If there is any way to solve this surface area or area without integrating, that would be preferential, as I don't think this problem is supposed to make us get into integration. But any thoughts are much appreciated. Thanks!
 

pray4mojo

Diamond Member
Mar 8, 2003
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Originally posted by: SnoPearL69
Okay, upon further thought of the problem, this can either be solved in 3D (sphere), or approximated in 2D (circle of radius r).

In 3D, I'm talking about a cut of distance h from the surface, towards the center (along a radius). So, imagine sawing off the skull cap of someone's head, and you're cutting a distance of h from the very top towards the center. I need the surface area of that.

In 2D, it's similar, except I'm looking for the area of a section of the circle, again a distance h from the border towards the center, along a radius. For this, imagine a sunrise, and the portion of the sun that has appeared over the horizon is a length of h from top of the arch to the plane of the horizon.

If there is any way to solve this surface area or area without integrating, that would be preferential, as I don't think this problem is supposed to make us get into integration. But any thoughts are much appreciated. Thanks!

The first thing that came into my head was integration although I don't know how to do it. I'd like to see this done without integration...
 

Justin218

Platinum Member
Jan 21, 2001
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I'm pretty sure you use multivarible calculus to do that. not sure if you can without.
 

Lorax

Golden Member
Apr 14, 2000
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all i came up with was (4*pi*r^2)/2 for the whole half that wasn't sliced, then maybe take a proportion, but i don't think you can do it without calculus.
 

eigen

Diamond Member
Nov 19, 2003
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Originally posted by: Heisenberg
I still don't really understand the question. Can you rephrase it? I would just integrate over the surface using the appropriate limits for the section you want.

This is right.Just dont intergrate around the whole thing.
 

LordMorpheus

Diamond Member
Aug 14, 2002
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Originally posted by: Heisenberg
I still don't really understand the question. Can you rephrase it? I would just integrate over the surface using the appropriate limits for the section you want.

bingo
 

TheLonelyPhoenix

Diamond Member
Feb 15, 2004
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Yeah, this is a multivariable question, you'd use a triple integral for this (one for each axis).

Don't remember how to set it up and I don't have a book lying around though, but maybe someone else does.