Math Is Fun Forum

bossk171

Member

Registered: 2007-07-16

Posts: 305

Volume of a Ring

I've gone over this a few times and I'm getting the same answer ever time. I don't believe my answer.

Consider a semi circle with radius "r" centered about the y axis and distance "k" from the origin. Put another way, consider the region enclosed by the curves:

and

Now rotate this region around the x-axis and you'll end up with a ring. What is the volume of this ring?

The answer I'm getting is

Which is ridiculous, not only is it independent of the distance "k" from the x axis (the radius of the ring hole), but it's the equation of the volume of a sphere.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

juriguen

Member

Registered: 2009-07-05

Posts: 59

Re: Volume of a Ring

Hi!

First of all, I think your equations do not define a ring, but a sphere... At least you write nothing about being a hole centered at k and of any radius.

Anyway, the following website has the solution and, apparently, you got close to the solution!
http://plus.maths.org/issue1/puzzle/index.html

Hope it helps
Jose

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“Make everything as simple as possible, but not simpler.” -- Albert Einstein

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Volume of a Ring

Are you describing a torus?

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Why did the vector cross the road?
It wanted to be normal.

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Volume of a Ring

No, I'm not defining a torus. A torus is a circle rotated around an axis, this is a semi circle rotated around an axis. It's close to a torus, but with a cylindrical whole.

Juriguen, that problem is similar to mine, but not the same. for your problem, there is a sphere with a cylindrical hole cut into it. Mine is not spherical.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

Fruityloop

Member

Registered: 2009-05-18

Posts: 143

Re: Volume of a Ring

I am not 100% sure of myself but after a rather lengthy process I got...

I used volume by slicing using

as the radius of the disks and then subtracting out the middle cylinder which has a radius of k and a height of 2r.

Last edited by Fruityloop (2009-08-15 16:11:28)

Fruityloop

Member

Registered: 2009-05-18

Posts: 143

Re: Volume of a Ring

I got the same answer by a different method. I first calculated the centroid of a semi-circle of radius r centered on the y-axis.
the x-coordinate of the centroid will be zero, so we just need to calculate the y-coordinate...

which comes out to be

now we need to divide by the area so...

so the centroid is located at

The volume is equal to the area multiplied by the distance traveled by the centroid. So we have..

Last edited by Fruityloop (2011-08-09 00:11:53)