Problem 23 - Calculus and its applications

Question

A triple iterated integral such as $$\int_{r}^{s} \int_{c}^{d} \int_{a}^{b} f(x, y, z) d x d y d z$$ is evaluated in much the same way as a double iterated integral. We first evaluate the inside \(x\) -integral, treating \(y\) and \(z\) as constants. Then we evaluate the middle y-integral, treating \(z\) as \(a\) constant. Finally, we evaluate the outside z-integral. Evaluate the following triple integrals. $$ \int_{0}^{1} \int_{1}^{3} \int_{-1}^{2}(2 x+3 y-z) d x d y d z $$

Step-by-Step Solution

Answer

The value of the triple integral is 23.

Step 1: Evaluate the innermost integral with respect to x

Evaluate the integral \( \int_{-1}^{2} (2x + 3y - z) \ dx \). Consider \( y \) and \( z \) as constants during this process. This simplifies to \( \left[ x^2 + 3xy - xz \right]_{-1}^{2} \). Calculate by plugging in the limits for \( x \):\[ \left[ (2)^2 + 3(2)y - 2z \right] - \left[ (-1)^2 + 3(-1)y - (-1)z \right] = \left[ 4 + 6y - 2z \right] - \left[ 1 - 3y + z \right] \]After simplification, the result is \( 9y - 4z + 3 \).

Step 2: Evaluate the middle integral with respect to y

Now integrate \( \int_{1}^{3} (9y - 4z + 3) \ dy \), treating \( z \) as a constant. The integral is \( \left[ \frac{9}{2}y^2 - 4yz + 3y \right]_{1}^{3} \). Calculate by substituting the limits for \( y \):\[ \left[ \frac{9}{2}(3)^2 - 4(3)z + 3(3) \right] - \left[ \frac{9}{2}(1)^2 - 4z + 3 \right] = \left[ \frac{81}{2} - 12z + 9 \right] - \left[ \frac{9}{2} - 4z + 3 \right] \]After simplifying, this results in \( 27 - 8z \).

Step 3: Evaluate the outermost integral with respect to z

Integrate \( \int_{0}^{1} (27 - 8z) \ dz \). This integral simplifies to \( \left[ 27z - 4z^2 \right]_{0}^{1} \). Plugging in the limits for \( z \):\[ \left[ 27(1) - 4(1)^2 \right] - \left[ 27(0) - 4(0)^2 \right] = 27 - 4 \]Thus, the final value is \( 23 \).

Key Concepts

Iterated Integrals

Triple integrals are an extension of double integrals, allowing us to explore and measure volumes or other quantities over three-dimensional spaces. When faced with a triple integral, we often turn to the method of iterated integrals, approaching the problem by integrating step-by-step across each variable.

• Begin with the innermost integral, treating the other variables as constants.
• Move outward to each subsequent integral, applying similar logic for each variable.
• This structured method simplifies the complexity of integrating over multiple dimensions.

By breaking the process into smaller, manageable parts, iterated integrals guide us through complex calculus challenges, providing a clear path to solve for the entire volume or entity specified by the integral limits.

Integration Steps

Evaluating a triple integral involves a sequential process much like peeling an onion, where each layer reveals more information. Each step in the integration process is crucial for accuracy.

**Step-by-Step Assessment:**
- **Innermost Integral:** Start by integrating with respect to the first variable, assuming other variables are constants. - **Middle Integral:** Next, proceed with the integration for the second variable, which provides a new expression that depends only on the remaining variable(s). - **Outermost Integral:** Finally, integrate the resulting function with respect to the last variable. By following this orderly sequence, we can successfully determine the function or value defined by the triple integral limits.

Calculus Problem Solving

Triple integrals, while more complex, are merely an extension of understanding foundational calculus concepts and applying them to higher-dimensional problems. Effective problem-solving in calculus requires a strategic approach and keen attention to detail.
**Key Problem-Solving Tips:**

• Understand the function and its domain, especially when dealing with complex expressions.
• Carefully set integration limits to reflect accurately the region being analyzed.
• Break the problem into smaller tasks—integrate iteratively to simplify the process.
• Check work at each integration step to ensure accuracy before moving forward.

By mastering these calculus problem-solving techniques, students can tackle triple integral challenges with confidence, ensuring a deeper comprehension of multi-variable calculus.