First Pappus-Guldinus Theorem

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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.
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Also known as

The first Pappus-Guldinus theorem 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

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