Essential Concepts

• Finding the mass, center of mass, moments, and moments of inertia in double integrals:
  • For a lamina $R$ with a density function $\rho(x,y)$ at any point $(x,y)$ in the plane, the mass is $m=\underset{R}{\displaystyle\iint} \rho(x,y) dA$
  • The moments about the $x$-axis and $y$-axis are $M_{x}=\underset{R}{\displaystyle\iint} y\rho(x,y) dA$ and $M_{y}=\underset{R}{\displaystyle\iint} x\rho(x,y) dA$
  • The center of mass is given by $\overline{x}=\frac{M_{y}}{m}$, $\overline{y}=\frac{M_{x}}{m}$
  • The center of mass becomes the centroid of the plane when the density is constant.
  • The moments of inertia about the $x$-axis, $y$-axis, and the origin are $I_{x}=\displaystyle\iint_{R} y^{2}\rho(x,y) dA$, $I_{y}=\underset{R}{\displaystyle\iint} x^{2}\rho(x,y) dA$, and $I_{0}=I_{x}+I_{y}=\underset{R}{\displaystyle\iint} (x^{2}+y^{2})\rho(x,y) dA$
• Finding the mass, center of mass, moments, and moments of inertia in triple integrals:
  • For a solid object $Q$ with a density function $\rho(x,y,z)$ at any point $(x,y,z)$ in space, the mass is $m=\displaystyle\iiint_{Q} \rho(x,y,z) dV$
  • The moments about the $xy$-plane, the $xz$-plane, and the $yz$-plane are $M_{xy}=\displaystyle\iiint_{Q} z\rho(x,y,z) dV$, $M_{xz}=\underset{Q}{\displaystyle\iiint} y\rho(x,y,z) dV$, $M_{yz}=\underset{Q}{\displaystyle\iiint} x\rho(x,y,z) dV$
  • The center of mass is given by $\overline{x}=\frac{M_{yz}}{m}$, $\overline{x}=\frac{M_{xz}}{m}$, $\overline{x}=\frac{M_{xy}}{m}$
  • The center of mass becomes the centroid of the solid when the density is constant.
  • The moments of inertia about the $yz$-plane, the $xz$-plane, and the $xy$-plane are $I_{x}=\underset{Q}{\displaystyle\iiint} (y^{2}+z^{2})\rho(x,y,z) dV$, $I_{y}=\underset{Q}{\displaystyle\iiint} (x^{2}+z^{2})\rho(x,y,z) dV$, $I_{z}=\underset{Q}{\displaystyle\iiint} (x^{2}+y^{2})\rho(x,y,z) dV$