Solid of revolution knowing only area?

1. Jan 28, 2013

joeblow

Suppose I have a region R whose boundary extremely complicated. While it would take me hundreds of years to approximate the boundary with formulae, I can easily estimate the area of R within a desired precision. I want to find the volume of the solid of revolution of R .

My intuition told me that I should be able to integrate some function solely dependent on the area with respect to θ from 0 to 2π .

However, using what knowledge I already have, this is not the case. For instance, the volume of the solid obtained by rotating a right triangle whose base is r and height is h about the axis coninciding with the leg of length h is given by the well-known formula $$\frac{1}{3}\pi r^{2}h=\pi r(\frac{1}{2}rh)\cdot\frac{2}{3}=\frac{2A\pi r}{3}$$. The volume of a solid of a similar solid formed by a rectangle of base r and height 1/2 h is given by the well-known fromula $$\pi r^{2}(\frac{1}{2}h)=\pi r(\frac{1}{2}rh)=A\pi r$$. The solids are both generated by regions of equal area, yet the volumes are different by a factor of 2/3 .

My question is what parameters are involved in this process? Clearly, I am missing at least one.

2. Jan 28, 2013

Staff: Mentor

It is not sufficient to know the area - you have to know how far away this area is from the rotation axis "on average" (as integral).

3. Jan 28, 2013

joeblow

How would the centroid of the region figure into my volume? I am picturing a small portion of the solid subtending an angle of dθ.

4. Jan 28, 2013

Staff: Mentor

That is not enough. If you can calculate the average x (which leads to the barycenter) and the average x^2 (in physics, this would be related to the inertial moment) for your area, this might work. x is the distance to the rotation axis here.

5. Jan 28, 2013

rollingstein

Pappus' 2nd centroid theorem

The second theorem states that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by its geometric centroid.

http://en.wikipedia.org/wiki/Pappus's_centroid_theorem

6. Jan 28, 2013

joeblow

Bingo. Thanks.

7. Jan 29, 2013

joeblow

There's no proof provided in that page, but you can prove it by doing the "natural" process. That is, imagine the solid as a layered cake whose layers are all Δy tall. Take one particular slice subtending an angle of Δθ. Then, you estimate the volume of a slice by adding the volumes of Δθ-slices of cylinders of height Δy and radius x (dependent on y, so it's different at each layer). Taking the limit as Δy→0, you get an integral that you can simplify using the properties of the x-coordinate of the center of mass. Then, just add up all the slices.

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