First Pappus-Guldinus Theorem

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $C$ be a plane figure that lies entirely on one side of a straight line $\LL$.

Let $S$ be the solid of revolution generated by $C$ around $\LL$.


Then the volume of $S$ is equal to the area of $C$ multiplied by the distance travelled by the centroid of $C$ around $\LL$ when generating $S$.


Proof

Let $V$ denote the volume of $S$

Let $\AA$ denote the area of $C$.

Let $C$ be embedded in a Cartesian plane such that $\LL$ coincides with the $x$-axis.

Let $\tuple {\overline x, \overline y}$ be the coordinates of the centroid of $C$.

Consider a rectangle $R$ bounded by the lines:

$y = 0$
$x = \xi$
$x = \xi + \delta x$
$y = \map f x$

The moment $M_y$ of $R$ about the $y$-axis is given by:

$M_y = \map f x \xi \rdelta \xi$

Hence from Area under Curve:

$\AA \overline x = \ds \int_a^b x \map f x \rd x$

The moment $M_x$ of $R$ about the $x$-axis is given by:

$M_x = y \rdelta x \dfrac y 2$

that is, half way up.

Hence:

$\AA \overline y = \dfrac 1 2 \ds \int_a^b y^2 \rd x$

It follows immediately that:

$\dfrac V \AA \overline y = 2 \pi$

That is:

$V = 2 \pi \AA \overline y$


The validity of the material on this page is questionable.
In particular: The above is the exposition given in L. Harwood Clarke: A Note Book in Pure Mathematics. It is based on an unstated assumption that $C$ is bounded on one side by the $x$ axis, which is not necessarily the case. I will get round to hunting down a rigorous exposition of this. I must have it on my bookshelf somewhere in one of my mechanics texts. If anyone wants to take this on and make it rigorous, feel free to do so.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Questionable}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.



Also known as

This result is also known as:

Pappus's Centroid Theorem for Volume
the First Guldinus Theorem.


Also see


Source of Name

This entry was named for Pappus of Alexandria and Paul Guldin.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=First_Pappus-Guldinus_Theorem&oldid=603989"