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To calculate the triple integral of the function \( f(x, y, z) = xyz^2 \) over the rectangular box \( B \) given by:

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

we will integrate the function with respect to \( z \) first, then \( y \), and finally \( x \).

### Step 1: Integrating with respect to \( z \)

First, integrate the function with respect to \( z \) while treating \( x \) and \( y \) as constants:

\[
\int_0^6 xyz^2 \, dz = xy \int_0^6 z^2 \, dz = xy \left[ \frac{z^3}{3} \right]_0^6 = xy \left( \frac{6^3}{3} \right) = 72xy
\]

### Step 2: Integrating with respect to \( y \)

Next, integrate the result from Step 1 with respect to \( y \):

\[
\int_{-2}^4 72xy \, dy = 72x \int_{-2}^4 y \, dy = 72x \left[ \frac{y^2}{2} \

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 solution calculates the triple integral ∭_B xyz^2 dV, where the box B is defined by the region 0 ≤ x ≤ 2, -2 ≤ y ≤ 4, and 0 ≤ z ≤ 6. It involves step-by-step integration with respect to z, y, and x using Fubini's Theorem.

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