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

Oscar Ramos

Oscar Ramos

Answered question

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?

Answer & Explanation

Hayden Dudley

Hayden Dudley

Beginner2023-03-04Added 6 answers

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 π 4 the line x + y = c must divide the quarter circle into 2 pieces each with area π 8 .
The area of the triangle formed by the X-axis, the Y-axis, and x + y = c is c 2 2
Thus
c 2 2 = π 8
c = π 2
and the centroid is located at the midpoint of the line segment
( π 4 , π 4 )

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