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

1 Answer
Mar 11, 2015

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).
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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)#