Solved: (Center of gravity) Justity the following formulas
Chapter 10, Problem 10.1.126(choose chapter or problem)
Justify the following formulas for the mass M and the center of gravity \((\bar{x}, \bar{y}, \bar{z})\) of a lamina S of density (mass per unit area) \(\sigma(x, y, z)\) in space:
\(M=\iint_{S} \sigma d A, \quad \bar{x}=\frac{1}{M} \int_{S} x \sigma d A\)
\(\bar{y}=\frac{1}{M} \iint_{S} y \sigma d A, \quad \bar{z}=\frac{1}{M} \iint_{S} z \sigma d A\)
Text Transcription:
(bar x, bar y, bar z)
sigma(x, y, z)
M = iint_S sigma dA, bar x = 1 / M int_S x sigma dA
bar y = 1 / M iint_S y sigma dA, bar z = 1 / M iint_S z sigma dA
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