The value of the definite integral $$\int_{-3}^{3} \int_{-2}^{2} \int_{-1}^{1} (4x^2y - z^3) \, dz \, dy \, dx$$ is _________ . (Rounded off to the nearest integer)

We are asked to evaluate the triple integral:

By linearity, the integral can be split into two separate integrals:

where:

Since the integration limits are independent constants, each multiple integral can be factored into single variable integrals:

The function $y$ is an odd function, and the limits of integration $[-2, 2]$ are symmetric about zero:

Thus, $I_1 = 0$.

Similarly, for $I_2$:

Since $z^3$ is also an odd function over the symmetric interval $[-1, 1]$:

Thus, $I_2 = 0$.

Therefore: