Question

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

Answer 1

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 `pi/4`
the line `x+y=c` must divide the quarter circle into `2` pieces each with area `pi/8`.

The area of the triangle formed by the X-axis, the Y-axis, and `x+y=c`
is `(c^2)/2`

Therefore
`(c^2)/2 = pi/8`
`rarr c = (sqrt(pi))/2`

and the centroid is located at the midpoint of the line segment
`( (sqrt(pi))/4, (sqrt(pi))/4)`