CalcRef

Applications of Integrals · Centroids & Averages

Centroid (y-coordinate)

yˉ=1A∫ab12([f(x)]2−[g(x)]2)dx\bar{y} = \frac{1}{A}\int_a^b \frac{1}{2}\left([f(x)]^2 - [g(x)]^2\right) dx

The y-coordinate of the centroid of the region between f(x) and g(x).

Variables

SymbolNameUnit
aLeft bound—
bRight bound—

Worked examples

Find ȳ for the region under y = x² on [0, 2].
  1. A = 8/3
  2. ȳ = (3/8)(1/2) ∫₀² (x²)² dx = (3/16) ∫₀² x⁴ dx = (3/16)(32/5) = 6/5

Answer: ȳ = 6/5

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