CalcRef

Applications of Integrals · Centroids & Averages

Centroid (x-coordinate)

xˉ=1A∫abx[f(x)−g(x)] dx\bar{x} = \frac{1}{A}\int_a^b x[f(x) - g(x)]\, dx

The x-coordinate of the centroid of the region between f(x) and g(x). A is the area of the region.

Variables

SymbolNameUnit
aLeft bound—
bRight bound—

Worked examples

Find x̄ for the region under y = x² on [0, 2].
  1. A = ∫₀² x² dx = 8/3
  2. x̄ = (3/8) ∫₀² x · x² dx = (3/8) ∫₀² x³ dx = (3/8)(4) = 3/2

Answer: x̄ = 3/2

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