MvCalc57

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Mass of the tetrahedron=\iiint _{V}\rho dxdydz\, where rho is the constant density of the substance.

M=\rho \int _{0}^{a}\int _{{y=0}}^{{b(1-{\frac  {x}{a}})}}\int _{{z=0}}^{{c(1-{\frac  {x}{a}}-{\frac  {y}{b}})}}dzdydx\,

=\rho \int _{0}^{a}\int _{{y=0}}^{{b(1-{\frac  {x}{a}})}}c(1-{\frac  {x}{a}}-{\frac  {y}{b}})dydx\,

=\rho c\int _{0}^{a}[(1-{\frac  {x}{a}})y-{\frac  {y^{2}}{2b}}]_{{0}}^{{b(1-{\frac  {x}{a}})}}dx\,

={\frac  {cb\rho }{2}}\int _{0}^{a}(1-{\frac  {x}{a}})^{2}dx={\frac  {\rho bc}{2}}{\frac  {a}{3}}={\frac  {\rho abc}{6}}\,

Let x1,y1,z1 be the coordinates of the centroid.Then

Mx_{1}=\rho \int _{0}^{a}\int _{{y=0}}^{{b(1-{\frac  {x}{a}})}}\int _{{z=0}}^{{c(1-{\frac  {x}{a}}-{\frac  {y}{b}})}}xdzdydx\,

=\rho \int _{0}^{a}\int _{{y=0}}^{{b(1-{\frac  {x}{a}})}}cx(1-{\frac  {x}{a}}-{\frac  {y}{b}})dydx\,

=c\rho \int _{0}^{a}[x(1-{\frac  {x}{a}})y-{\frac  {xy^{2}}{ab}}]_{{y=0}}^{{b(1-{\frac  {x}{a}})}}dx\,

=c\rho b\int _{0}^{a}x(1-{\frac  {x}{a}})^{2}dx=\rho bc{\frac  {a^{2}}{12}}\,

x_{1}={\frac  {\rho a^{2}bc}{12}}{\frac  {6}{\rho abc}}={\frac  {a}{4}}\,

Similarly,y_{1}={\frac  {b}{4}},z_{1}={\frac  {c}{4}}\,

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