Solid of revolution knowing only area?

[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.

 

[mfb]

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).

 

[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.

 

[mfb]

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.

 

[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

 

[joeblow]

Bingo. Thanks.

 

[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.