The problem involves finding the center of mass for a tetrahedron $E$, which is bounded by the planes $x = 0$, $y = 0$, $z = 0$, and $x + y + z = 1$. The density function is $\rho(x, y, z) = y$. The goal is to find the center of mass for this solid.

Steps to Find the Center of Mass

1. Center of Mass Formula: The center of mass for a solid with a density function $\rho(x, y, z)$ is given by: $\left( \overline{x}, \overline{y}, \overline{z} \right) = \left( \frac{1}{M} \int \int \int_E x \rho(x, y, z) \, dV, \frac{1}{M} \int \int \int_E y \rho(x, y, z) \, dV, \frac{1}{M} \int \int \int_E z \rho(x, y, z) \, dV \right)$ where $M$ is the total mass of the solid: $M = \int \int \int_E \rho(x, y, z) \, dV$

2. Volume of the Tetrahedron: The region $E$ is the tetrahedron bounded by the planes $x = 0$, $y = 0$, $z = 0$, and $x + y + z = 1$. The vertices of the tetrahedron are: $(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)$ The volume of this tetrahedron can be computed using the formula: $\text{Volume} = \frac{1}{3!} \times \text{base area} \times \text{height}$ The volume is $\frac{1}{6}$.

3. Mass Calculation: The mass $M$ is given by the integral of the density function over the region $E$. Since $\rho(x, y, z) = y$, we integrate as follows: $M = \int_0^1 \int_0^{1-x} \int_0^{1-x-y} y \, dz \, dy \, dx$

4. Center of Mass Coordinates: To compute the coordinates of the center of mass, we will evaluate the integrals:

   • $\overline{x} = \frac{1}{M} \int_0^1 \int_0^{1-x} \int_0^{1-x-y} x \cdot y \, dz \, dy \, dx$
   • $\overline{y} = \frac{1}{M} \int_0^1 \int_0^{1-x} \int_0^{1-x-y} y^2 \, dz \, dy \, dx$
   • $\overline{z} = \frac{1}{M} \int_0^1 \int_0^{1-x} \int_0^{1-x-y} z \cdot y \, dz \, dy \, dx$

Let me proceed with the calculations now. sympy import symbols, integrate

Define the variables

x, y, z = symbols('x y z')

Define the density function

density = y

Define the limits for the region E (tetrahedron)

The region E is bounded by x + y + z =