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math question: partial surface area of sphere

S

SnoPearL69

Member
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)
 
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.
 
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...
 
Originally posted by: SnoPearL69
(Like, slicing a ball a distance h from the surface, and finding the surface area of that sliced off portion)
Click to expand...

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.

 
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!
 
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!
Click to expand...

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...
 
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.
 
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.
Click to expand...

This is right.Just dont intergrate around the whole thing.
 
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.
Click to expand...

bingo
 
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.
 
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