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Centroid x of area under y = 12x − 1.5x²

Short answer

Using vertical strip, area = 64 m², first moment = 160 m³, so centroidal x-coordinate = 2.5 m.

Determine centroidal x coordinate of the shaded area shown in figure below. [Figure: area under the curve y = 12x − 1.5x² above the x axis, from x = 0 to x = 4 m.]

Setting up: the area lies under y = 12x − 1.5x² from x = 0 to x = 4 m (y at x = 4 is 24 m). Take a vertical strip of height y and width dx, with x̃ = x.

Area

A = ∫₀⁴ (12x − 1.5x²) dx = [6x² − 0.5x³]₀⁴ = 6(16) − 0.5(64) = 96 − 32 = 64 m²

First moment about the y axis

∫ x dA = ∫₀⁴ x (12x − 1.5x²) dx = ∫₀⁴ (12x² − 1.5x³) dx = [4x³ − 0.375x⁴]₀⁴ = 4(64) − 0.375(256) = 256 − 96 = 160 m³

Centroid

x̄ = 160/64 = 2.5 m from the y axis.

✓ Answer checked by Claude Opus 5.5 (high effort)

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