Centroid, Area, and Moments of Inertia

Assume that an open or closed thin wire can be mathematically described by a vector-valued parametric curve r(t) = (x(t), y(t), z(t)) with t ∈ [a, b] and its density of mass is ρ(x, y, z). Then, • Length: L = ∫ b a √ x′(t)2 + y′(t)2 + z′(t)2dt. • Mass: M = ∫ b a ρ(x, y, z) √ x′(t)2 + y′(t)2 + z′(t)2dt. • Moment about the y-z plane: Myz = ∫ b a x(t)ρ(x, y, z) √ x′(t)2 + y′(t)2 + z′(t)2dt. • Moment about the x-z plane: Mxz = ∫ b a y(t)ρ(x, y, z) √ x′(t)2 + y′(t)2 + z′(t)2dt. • Moment about the x-y plane: Mxy = ∫ b a z(t)ρ(x, y, z) √ x′(t)2 + y′(t)2 + z′(t)2dt.