I ran into the following problem today:
- A region $R$ in the upper half-plane has center of mass at the point $(4,5)$, and area equal to $8$. Find the volume of the solid obtained by revolving $R$ about the $x$-axis.
I conjectured that, since the problem is given and an answer expected, perhaps this really is enough information. Therefore, we tried a couple of rectangles centered at $(4,5)$, and got the same answer: $80\pi$. I even have a proof, by calculation, that any rectangle meeting our criteria will give the same volume, and in fact I have a formula: $V=2\pi y_0A$, where $y_0$ and $A$ are the $y$-coordinate of the center of mass, and the area, respectively. I'm not sure how to move past rectangles.
Is it true, and if so how can we prove, that the volume in this scenario is invariant under transformations to $R$ that preserve area and the $y$-coordinate of the center of mass? (The $x$-coordinate clearly doesn't matter.)
Many thanks for any insight on this.
Best Answer
This is the Pappus-Guldin Theorem.
The volume is the area times the distance travelled by the centroid.
From a calculus point of view, it can be verified by looking at the formula for the volume of the solid of revolution (Method of Shells), and the formula for the ($y$-coordinate of the) centroid.