Triple Integral

Triple Integrals

A triple integral integrates over a 3D box $[x_a, x_b] \times [y_a, y_b] \times [z_a, z_b]$:

$\iiint_V f(x,y,z)\, dV$

The 3D Midpoint Rule

Divide each axis into $n$ equal intervals. Evaluate $f$ at each cell center:

$\iiint f\, dV \approx \sum_i \sum_j \sum_k f(x_i, y_j, z_k) \cdot \Delta x \cdot \Delta y \cdot \Delta z$

Geometric Meaning

• $\iiint 1\, dV$ = volume of the region
• $\iiint \rho(x,y,z)\, dV$ = total mass, if $\rho$ is density
• $\iiint f\, dV \;/\; \text{Volume}$ = average value of $f$ over the region

Examples

$\iiint 1\, dV$ over $[0,2] \times [0,3] \times [0,4]$ $= 24$ (volume of the box)

$\iiint x\, dV$ over $[0,1]^3$ $= \frac{1}{2} \cdot 1 \cdot 1 = \frac{1}{2}$

$\iiint (x+y+z)\, dV$ over $[0,1]^3$ $= \frac{3}{2}$ (by symmetry and linearity)

Applications

• Volume calculations of 3D regions
• Mass of non-uniform solid objects
• Center of mass calculations
• Moment of inertia

Your Task

Implement double triple_integral(double (*f)(double, double, double), double xa, double xb, double ya, double yb, double za, double zb, int n) using the 3D midpoint rule with $n$ divisions per axis.