How to find the centroid of the quarter circle of radius 1 with center at...

Non-Calculus Solution:
Observation 1:
The centroid must lie along the line ${\displaystyle y=x}$ (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 ${\displaystyle x+y=c}$ and furthermore, c must be less than 1 since the area of the triangle formed by the X-axis, Y-axis and ${\displaystyle x+y=1}$ is more than half of the area of the quarter circle.
Observation 3:
Since the area of the quarter circle (with radius = 1 is ${\displaystyle \frac{\pi }{4}}$ the line ${\displaystyle x+y=c}$ must divide the quarter circle into 2 pieces each with area ${\displaystyle \frac{\pi }{8}}$.
The area of the triangle formed by the X-axis, the Y-axis, and ${\displaystyle x+y=c}$ is ${\displaystyle \frac{c^2}{2}}$
Thus
${\displaystyle \frac{c^2}{2}=\frac{\pi }{8}}$
${\displaystyle \to c=\frac{\sqrt{\pi }}{2}}$
and the centroid is located at the midpoint of the line segment
${\displaystyle (\frac{\sqrt{\pi }}{4},\frac{\sqrt{\pi }}{4})}$