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The problem asks to compute the triple integral \( \iiint_B xyz^2 \, dV \), where \( B \) is a rectangular box (rectangular region) described by the following bounds:

\[
B = \{(x, y, z) \mid 0 \leq x \leq 2, -2 \leq y \leq 4, 0 \leq z \leq 6 \}
\]

The integral can be computed in the following steps:

1. **Set up the triple integral**:
   \[
   \iiint_B xyz^2 \, dV = \int_0^2 \int_{-2}^4 \int_0^6 xyz^2 \, dz \, dy \, dx
   \]

2. **Evaluate the innermost integral** (with respect to \( z \)):
   \[
   \int_0^6 xyz^2 \, dz = xy \int_0^6 z^2 \, dz
   \]
   The integral of \( z^2 \) from 0 to 6 is:
   \[
   \int_0^6 z^2 \, dz = \left[ \frac{z^3}{3} \right]_0^6 = \frac{6^3}{3} = \frac{216}{3} = 72
   \]
   So, the result of the innermost integral is:
   \[
   xy \cdot 72 = 72xy
   \]

3. **Evaluate the second integral** (with respect to \( y \)):
   \[
   \int_{-2}^4 72xy \, dy = 72x \int_{-2}^4 y \, dy
   \]
   The integral of \( y \) from \( -2 \) to \( 4 \) is:
   \[
   \int_{-2}^4 y \, dy = \left[ \frac{y^2}{2} \right]_{-2}^4 = \frac{4^2}{2} - \frac{(-2)^2}{2} = \frac{16}{2} - \frac{4}{2} = 8 - 2 = 6
   \]
   So, the result of the second integral is:
   \[
   72x \cdot 6 = 432x
   \]

4. **Evaluate the outermost integral** (with respect to \( x \)):
   \[
   \int_0^2 432x \, dx = 432 \int_0^2 x \, dx
   \]
   The integral of \( x \) from \( 0 \) to \( 2 \) is:
   \[
   \int_0^2 x \, dx = \left[ \frac{x^2}{2} \right]_0^2 = \frac{2^2}{2} = \frac{4}{2} = 2
   \]
   So, the result of the outermost integral is:
   \[
   432 \cdot 2 = 864
   \]

Thus, the value of the triple integral is \( \boxed{864} \).

---

Would you like further clarification or details on any step? Here are 5 related questions to deepen understanding:

1. What changes if the bounds for \( z \) are different?
2. How would you approach this problem if the integrand were \( xyz \) instead of \( xyz^2 \)?
3. Can this integral be solved using cylindrical or spherical coordinates?
4. What physical problems could be modeled by such a triple integral?
5. How does the result change if the order of integration is switched?

**Tip:** When solving triple integrals, make sure to clearly understand the limits for each variable based on the region of integration.

Assinale a alternativa que contenha o cálculo da integral tripla ∭_B xyz^2 dV, em que B é uma caixa retangular dada por: B = {(x,y,z) | 0 ≤ x ≤ 2, -2 ≤ y ≤ 4, 0 ≤ z ≤ 6}.

This problem involves the computation of a triple integral ∭_B xyz^2 dV over a rectangular box. The region B is bounded by 0 ≤ x ≤ 2, -2 ≤ y ≤ 4, and 0 ≤ z ≤ 6. Learn how to solve step-by-step using multivariable calculus and Fubini's Theorem.

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