Oscar Ramos

2023-03-03

How to find the centroid of the quarter circle of radius 1 with center at the origin lying in the first quadrant?

Hayden Dudley

Non-Calculus Solution:
Observation 1:
The centroid must lie along the line $y=x$ (otherwise the straight line running through (0,0) and the centroid would be to "heavy" on one side).
Observation 2:
For some constant, c, the centroid must lie along the line $x+y=c$ and furthermore, c must be less than 1 since the area of the triangle formed by the X-axis, Y-axis and $x+y=1$ is more than half of the area of the quarter circle.
Observation 3:
Since the area of the quarter circle (with radius = 1 is $\frac{\pi }{4}$ the line $x+y=c$ must divide the quarter circle into 2 pieces each with area $\frac{\pi }{8}$.
The area of the triangle formed by the X-axis, the Y-axis, and $x+y=c$ is $\frac{{c}^{2}}{2}$
Thus
$\frac{{c}^{2}}{2}=\frac{\pi }{8}$
$\to c=\frac{\sqrt{\pi }}{2}$
and the centroid is located at the midpoint of the line segment
$\left(\frac{\sqrt{\pi }}{4},\frac{\sqrt{\pi }}{4}\right)$

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