# Solid of revolution knowing only area?

• joeblow
In summary, the speaker has a complicated region R and wants to find the volume of the solid of revolution of R. They initially thought they could use an integral dependent on the area with respect to θ, but realized this was not the case. They question what parameters are involved in this process and mention Pappus' 2nd centroid theorem, which states that the volume of a solid of revolution is equal to the product of the area of the figure and the distance traveled by its centroid. They suggest using a layered cake analogy to find the volume through integration.
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.

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).

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.

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.

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

Bingo. Thanks.

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.

## 1. What is a solid of revolution?

A solid of revolution is a three-dimensional shape that is formed by rotating a two-dimensional shape around an axis. The resulting solid has the same cross-sectional area as the original shape, but with a larger or smaller volume depending on the axis of rotation.

## 2. How is the area of a solid of revolution calculated?

The area of a solid of revolution can be calculated using the formula A = πr2, where r is the radius of the cross-section of the solid. This formula is derived from the formula for the area of a circle.

## 3. Can the area of a solid of revolution be known without knowing its volume?

Yes, the area of a solid of revolution can be calculated without knowing its volume. This is because the area is only dependent on the shape and size of the cross-section, not the overall volume of the solid.

## 4. What information is needed to calculate the area of a solid of revolution?

To calculate the area of a solid of revolution, you need to know the shape and size of the cross-section, as well as the axis of rotation. This information can be used to determine the radius of the cross-section and the formula needed to calculate the area.

## 5. How is a solid of revolution used in real-world applications?

A solid of revolution is often used in engineering and design, particularly in the fields of architecture and manufacturing. It can also be used in mathematics to model real-world objects and understand their properties. For example, a solid of revolution can be used to model a water tower or a screwdriver handle.

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