Mr. Integral
Math notes/Calculus II/Geometry and Physical Applications

Work, Density and Center of Mass — Cheat sheet

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Core rule

W=∫Fdx,m=∫λdx,xˉ=∫xλdxm.W=\int Fdx,\quad m=\int\lambda dx,\quad \bar x=\frac{\int x\lambda dx}{m}.

Watch for

Use consistent distance units. Weight includes gravity; mass does not. A center of mass requires positive total mass.