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The problem you uploaded involves a triple integral:

\[
\int_0^3 \int_1^4 \int_1^2 dz \, dy \, dx
\]

Let's solve this step by step:

1. **First, integrate with respect to \( z \):**
   The limits of integration for \( z \) are from 1 to 2. Since there is no function of \( z \), the integral simply evaluates to the difference between the upper and lower limits:

   \[
   \int_1^2 dz = 2 - 1 = 1
   \]

2. **Second, integrate with respect to \( y \):**
   After evaluating the \( z \)-integral, we are left with:

   \[
   \int_1^4 1 \, dy = y \Big|_1^4 = 4 - 1 = 3
   \]

3. **Third, integrate with respect to \( x \):**
   Now, we integrate with respect to \( x \):

   \[
   \int_0^3 3 \, dx = 3x \Big|_0^3 = 3(3) - 3(0) = 9
   \]

Thus, the value of the triple integral is \( 9 \).

Do you need more details or explanations about any step? 

Here are 5 relative questions to expand your understanding:
1. What is a triple integral, and how is it used in calculus?
2. How do we decide the order of integration for a triple integral?
3. How would the result change if the limits of integration for \( z \) were different?
4. What is the geometric interpretation of triple integrals in three-dimensional space?
5. How would the process change if we were integrating a function instead of just a constant?

**Tip**: Always check the limits of integration and ensure that each integral is evaluated carefully in sequence.

Escolha a alternativa que contenha o resultado correto da integral tripla \( \int_0^3 \int_1^4 \int_1^2 dz \, dy \, dx \)

Solve the triple integral \( \int_0^3 \int_1^4 \int_1^2 dz \, dy \, dx \) with step-by-step solutions. Learn how to break down and calculate each part of the integral, understand key calculus concepts, and apply Fubini's theorem.

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