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To evaluate the triple integral \( \iiint_B x z^2 \, dV \) over the region \( B = [0,3] \times [6,8] \times [1,2] \), we can split the integral as follows:

1. Since \( B \) is a rectangular region, the integral becomes:
   \[
   \iiint_B x z^2 \, dV = \int_0^3 \int_6^8 \int_1^2 x z^2 \, dz \, dy \, dx
   \]

2. We first integrate with respect to \( z \), treating \( x \) and \( y \) as constants.

### Step 1: Integrate with respect to \( z \)
\[
\int_1^2 x z^2 \, dz = x \int_1^2 z^2 \, dz
\]
The integral of \( z^2 \) with respect to \( z \) is:
\[
\int z^2 \, dz = \frac{z^3}{3}
\]
Evaluating from \( z = 1 \) to \( z = 2 \):
\[
x \left[ \frac{z^3}{3} \right]_1^2 = x \left( \frac{2^3}{3} - \frac{1^3}{3} \right) = x \left( \frac{8}{3} - \frac{1}{3} \right) = x \cdot \frac{7}{3} = \frac{7x}{3}
\]

### Step 2: Integrate with respect to \( y \)
Now, the integral becomes:
\[
\int_0^3 \int_6^8 \frac{7x}{3} \, dy \, dx = \int_0^3 \frac{7x}{3} \left( \int_6^8 1 \, dy \right) dx
\]
Since \( \int_6^8 1 \, dy = 8 - 6 = 2 \), we have:
\[
\int_0^3 \frac{7x}{3} \cdot 2 \, dx = \int_0^3 \frac{14x}{3} \, dx
\]

### Step 3: Integrate with respect to \( x \)
Now we integrate \( \frac{14x}{3} \) with respect to \( x \):
\[
\int_0^3 \frac{14x}{3} \, dx = \frac{14}{3} \int_0^3 x \, dx
\]
The integral of \( x \) with respect to \( x \) is \( \frac{x^2}{2} \), so:
\[
\frac{14}{3} \left[ \frac{x^2}{2} \right]_0^3 = \frac{14}{3} \cdot \frac{3^2}{2} = \frac{14}{3} \cdot \frac{9}{2} = \frac{14 \cdot 9}{6} = \frac{126}{6} = 21
\]

### Final Answer:
\[
\iiint_B x z^2 \, dV = 21
\]

Would you like any additional details, or have questions?

---

1. What is a triple integral, and when is it used?
2. How does the order of integration affect the solution?
3. What are the steps for evaluating integrals over rectangular regions?
4. Can the result change if the region \( B \) changes?
5. Why is it useful to break down the integral step-by-step?

**Tip:** In rectangular regions, factor out constants to simplify integration step-by-step.

Evaluate \( \iiint_B x z^2 \, dV \), where the region \( B = [0,3] \times [6,8] \times [1,2] \).

This solution demonstrates the step-by-step evaluation of the triple integral \( \iiint_B x z^2 \, dV \) over the rectangular region \( B = [0,3] \times [6,8] \times [1,2] \). Suitable for students in advanced calculus, this solution covers concepts in multivariable calculus, including volume integration and the use of rectangular boundaries in triple integrals.

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