3D Integration
To find the total mass of a 3D object with density \(\rho(x, y, z)\), we use a Triple Integral:
\[M = \iiint_E \rho(x, y, z) dV = \int_{z_1}^{z_2} \int_{y_1(z)}^{y_2(z)} \int_{x_1(y,z)}^{x_2(y,z)} \rho(x, y, z) dx dy dz\]
Worked Examples
Example 1: Volume of a Unit Cube
Evaluate \(\int_0^1 \int_0^1 \int_0^1 1 dx dy dz\).
- Inner: \(\int_0^1 1 dx = 1\).
- Middle: \(\int_0^1 1 dy = 1\).
- Outer: \(\int_0^1 1 dz = 1\).
- Result: 1.
The Bridge to Quantum Mechanics
The "Expected Value" of any observable \(A\) in 3D is calculated using a triple integral: \(\langle A \rangle = \iiint \psi^* \hat{A} \psi dx dy dz\). This is the standard form of almost every equation in atomic physics. Whether we are calculating the magnetic moment of an atom or the probability of an electron being in a specific region of a crystal, we are performing triple integrals over Cartesian space.