Step-by-step explanation:
In the fourth quadrant, the equation of the unit circle is:
y = -√(1 − x²), 0 ≤ x ≤ 1
The x and y coordinates of the centroid are:
cₓ = (∫ x dA) / A = (∫ xy dx) / A
cᵧ = (∫ y dA) / A = (∫ ½ y² dx) / A
For a quarter circle in the fourth quadrant, A = -π/4.
Solving each integral:
∫₀¹ xy dx
= ∫₀¹ -x √(1 − x²) dx
= ½ ∫₀¹ -2x √(1 − x²) dx
If u = 1 − x², then du = -2x dx.
When x = 0, u = 1. When x = 1, u = 0.
= ½ ∫₁⁰ √u du
= ½ ∫₁⁰ u^½ du
= ½ (⅔ u^³/₂) |₁⁰
= (⅓ u√u) |₁⁰
= 0 − ⅓
= -⅓
∫₀¹ ½ y² dx
= ½ ∫₀¹ (1 − x²) dx
= ½ (x − ⅓ x³) |₀¹
= ½ [(1 − ⅓) − (0 − 0)]
= ⅓
Therefore, the x and y coordinates of the centroid are:
cₓ = (-⅓) / (-π/4) = 4/(3π)
cᵧ = (⅓) / (-π/4) = -4/(3π)
