1. **State the problem:** Evaluate the triple integral $$\iiint 12xy^2 z^3 \, dV$$ over the given volume (note: the volume or limits are not specified, so we assume the problem is to set up or simplify the integrand for evaluation). 2. **Understand the integral:** The integrand is $$12xy^2 z^3$$, which is a product of variables raised to powers multiplied by a constant. 3. **If limits were given, we would integrate step-by-step:** - Integrate with respect to one variable while treating others as constants. - Use the power rule for integration: $$\int x^n dx = \frac{x^{n+1}}{n+1}$$. 4. **Since no limits or volume are specified, the integral cannot be evaluated numerically.** 5. **Summary:** To evaluate, you need the limits of integration or the volume over which to integrate. Without these, the integral remains $$\iiint 12xy^2 z^3 \, dV$$. If you provide the limits or the region, I can help evaluate it step-by-step.