To solve the integral $\int_0^4 3x^2 \, dx$, we proceed as follows:

1. Antiderivative: First, find the antiderivative of $3x^2$. $\int 3x^2 \, dx = 3 \cdot \frac{x^3}{3} = x^3$

2. Apply limits of integration: Now, apply the limits of integration from 0 to 4. $\left[ x^3 \right]_0^4 = 4^3 - 0^3 = 64 - 0 = 64$

So, the value of the integral is $\boxed{64}$.

Would you like more details or have any questions?

Here are some related questions to deepen understanding:

1. What is the general process for finding definite integrals of polynomial functions?
2. How does the power rule for integration work?
3. How do we handle integrals involving negative powers of $x$?
4. Can this result be visualized geometrically, and what would it represent?
5. How does the Fundamental Theorem of Calculus apply to this problem?

Tip: Always double-check if you're applying the correct limits of integration when solving definite integrals!