Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate the iterated integrals. \(\int_{0}^{5} \int_{0}^{3} \int_{z^{2}}^{9} x y z d x d z d y\)

Step-by-Step Solution

Answer

The value of the iterated integral is 1518.75.

Step 1: Evaluate Inner Integral

The inner integral is with respect to \(x\) and is given by \(\int_{z^{2}}^{9} x y z \, dx\). Since \(y\) and \(z\) are treated as constants, integrating \(x y z\) with respect to \(x\) results in \[ \frac{x^2}{2} y z \bigg|_{z^2}^{9} = \frac{81}{2} y z - \frac{z^4}{2} y z. \] Simplifying gives \( \left( \frac{81}{2} - \frac{z^4}{2} \right) yz \).

Step 2: Evaluate Middle Integral

Next, integrate the result from Step 1 with respect to \(z\) over the interval \([0, 3]\). The integral becomes \[ \int_{0}^{3} \left( \frac{81}{2} - \frac{z^4}{2} \right) y z \, dz. \] This can be divided into two separate integrals: \( \int_{0}^{3} \frac{81}{2} y z \, dz \) and \( -\int_{0}^{3} \frac{z^5}{2} y \, dz \). Evaluate each integral: \[ \frac{81}{2} y \left( \frac{z^2}{2} \right) \bigg|_{0}^{3} = \frac{81}{2} y \cdot \frac{9}{2} = \frac{729}{4} y, \] and \[ -\frac{y}{2} \cdot \frac{z^6}{6} \bigg|_{0}^{3} = -\frac{y}{2} \cdot \frac{729}{6} = -\frac{243}{4} y. \] Adding these results, we get \( \frac{486}{4} y = \frac{243}{2} y \).

Step 3: Evaluate Outer Integral

Finally, integrate the result from Step 2 with respect to \(y\) over the interval \([0, 5]\): \( \int_{0}^{5} \frac{243}{2} y \, dy \). This gives: \[ \frac{243}{2} \cdot \frac{y^2}{2} \bigg|_{0}^{5} = \frac{243}{2} \cdot \frac{25}{2} = \frac{243 \times 25}{4} = 1518.75. \]

Step 4: Conclusion

Through a series of evaluations from the innermost to the outermost integral, we determined the overall value of the iterated integral. This involved careful integration with respect to each variable in turn.

Key Concepts

Triple Integration

Triple integration is the process of evaluating expressions involving three variables through integration. This method efficiently computes volumes and averages over three-dimensional regions. By handling one variable at a time from the innermost to the outermost integral, we simplify complex problems. In the given exercise, we performed triple integration to evaluate the iterated integral

• First, we integrated with respect to the variable \(x\).
• Then, moved to \(z\), and,
• Finally, with respect to \(y\).

This systematic approach helps avoid errors and breaks down tasks into manageable steps. Remember, each integration treats other variables as constants until their respective turn. This ensures smooth computation.

Multiple Integrals

Multiple integrals extend the concept of integration beyond single-variable calculus, accommodating functions of two or more variables. This allows for the computation of areas, volumes, or even higher-dimensional constructs. In our exercise, we dealt with a triple integral, which involves three layers of integration. It began with integrating over a specific range of one variable at a time.

• The process reflects integration over multiple dimensions, essential for exploring physical phenomena.
• By evaluating these iterated integrals, we identified the collective impact of all variables, maintaining the chain of integration within bounds.

Techniques applied in multiple integrals can scale up to accommodate various real-world applications.

Evaluation of Integrals

When evaluating integrals, especially iterated ones, strategic execution is key. Our specific exercise required us to simplify expressions step by step to reach an accurate solution.

• First, we focused on calculating the inner integral concerning \(x\).
• Then, proceeded outward towards \(z\) and finally \(y\), each integration simplifying the problem further down to a numerical value.

Effective evaluation involves:

• Recognizing which variables to hold constant,
• Splitting complex integrals into simpler parts,
• Accurately applying known integral formulas.

Following these principles ensures precision across potentially intricate calculations inherent in triple integrals.

Calculus Techniques

Mastering calculus techniques is critical for solving multiple integrals. In this instance, key methodologies were:

• Breaking down the integration process into a sequence of single-variable integrals,
• Simplifying each step through methodical algebraic manipulation,
• Applying calculus rules for changing integration order or simplifying convoluted expressions.

Emphasizing clarity at every stage is crucial. Calculus becomes less daunting by understanding each action's purpose and logically transitioning between steps. For instance, transitioning smoothly from integrating with respect to \(x\) to \(z\), and finally \(y\), capitalizes on learned techniques to solve complex multi-variable calculus problems smoothly and accurately.