Can the centroid of a semicircle be computed without deferring to calculus or a limiting procedure?
[Math] Compute the Centroid of a Semicircle without Calculus
geometry
geometry
Can the centroid of a semicircle be computed without deferring to calculus or a limiting procedure?
Best Answer
The following may be acceptable to you as an answer. You can use the centroid theorem of Pappus.
I do not know whether you really mean half-circle (a semi-circular piece of wire), or a half-disk. Either problem can be solved using the theorem of Pappus.
When a region is rotated about an axis that does not go through the region, the volume of the solid generated is the area of the region times the distance travelled by the centroid.
A similar result holds when a piece of wire is rotated. The surface area of the solid is the length of the wire times the distance travelled by the centroid.
In the case of rotating a semi-circular disk, or a semi-circular piece of wire, the volume (area) are known.
Remark: The result was known some $1500$ years before Newton was born. And the volume of a sphere, also the surface area, were known even before that. The ideas used to calculate volume, area have, in hindsignt, limiting processes at their heart. So if one takes a broad view of the meaning of "calculus," we have not avoided it.