The image contains mathematical formulas related to calculating the center of mass of a three-dimensional object with a given density function $\rho(x, y, z)$. Here is a detailed breakdown of the provided formulas:

1. Total Mass of the Object: The mass $M$ of the volume $V$ is given by the triple integral: $M = \iiint_V \rho(x, y, z) \, dx \, dy \, dz$

   This computes the total mass by integrating the density function $\rho(x, y, z)$ over the entire volume $V$.

2. Center of Mass Formulas: The coordinates of the center of mass $(\bar{x}, \bar{y}, \bar{z})$ are defined as: $\bar{x} = \frac{1}{M} \iiint_V x \, \rho(x, y, z) \, dx \, dy \, dz$ $\bar{y} = \frac{1}{M} \iiint_V y \, \rho(x, y, z) \, dx \, dy \, dz$ $\bar{z} = \frac{1}{M} \iiint_V z \, \rho(x, y, z) \, dx \, dy \, dz$

   • $\bar{x}$: Weighted average of $x$ values over the volume, with $\rho(x, y, z)$ acting as the weight.
   • $\bar{y}$: Weighted average of $y$ values.
   • $\bar{z}$: Weighted average of $z$ values.

3. Summary:
   • $\iiint_V$: Denotes a triple integral over the 3D volume $V$.
   • $dx \, dy \, dz$: Differential volume element in Cartesian coordinates.
   • The density function $\rho(x, y, z)$ determines how the mass is distributed throughout the volume.

Do you need a worked-out example or further clarifications on these formulas?

Here are some questions to expand the discussion:

1. What is the physical interpretation of the center of mass in a system?
2. How do the formulas change if the density $\rho$ is constant?
3. Can these formulas be applied in cylindrical or spherical coordinates? How?
4. What are some practical applications of these formulas in engineering or physics?
5. How would you set up these integrals for a specific volume, such as a sphere or a cube?

Tip:

To simplify computation, exploit symmetry in $\rho(x, y, z)$ or $V$ when calculating the center of mass. Symmetry can often make some terms of the integral equal to zero.