Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1–10, evaluate the iterated integrals. $$ \int_{0}^{2} \int_{-1}^{4} \int_{0}^{3 y+x} d z d y d x $$

Step-by-Step Solution

Answer

The value of the iterated integral is 55.

Step 1: Understand the Order of Integration

The integrals are nested, starting from the innermost integral to the outermost. Here, the order of integration is: integrate with respect to \(z\), then \(y\), and finally \(x\).

Step 2: Evaluate the Innermost Integral

The innermost integral is \(\int_{0}^{3y+x} dz\). Since the integral is with respect to \(z\) and there is no \(z\)-dependency, it simplifies to the upper limit of integration: \[\int_{0}^{3y+x} dz = [z]_{0}^{3y+x} = 3y+x\]

Step 3: Substitute and Simplify

Substitute the result from the innermost integral back into the middle integral. The integrand becomes \( \int_{-1}^{4} (3y+x) dy \).

Step 4: Evaluate the Middle Integral

Calculate the integral \(\int_{-1}^{4} (3y+x) dy\). This simplifies to: \[\int_{-1}^{4} 3y \, dy + \int_{-1}^{4} x \, dy\]The first part:\[\int_{-1}^{4} 3y \, dy = \left[\frac{3y^2}{2}\right]_{-1}^{4} = \frac{3(16) - 3(1)}{2} = \frac{45}{2}\]The second part:\[\int_{-1}^{4} x \, dy = x[y]_{-1}^{4} = 5x\]Thus, the middle integral result is \( \frac{45}{2} + 5x \).

Step 5: Substitute and Integrate the Outermost Integral

Substitute the result from the middle integral back for the outermost integral to get \(\int_{0}^{2} \left( \frac{45}{2} + 5x \right) dx \). Evaluate:\[\int_{0}^{2} \frac{45}{2} \, dx + \int_{0}^{2} 5x \, dx\]First the constant term:\[\int_{0}^{2} \frac{45}{2} \, dx = \frac{45}{2}[x]_{0}^{2} = \frac{45}{2} \cdot 2 = 45\]Now the variable term:\[\int_{0}^{2} 5x \, dx = \left[\frac{5x^2}{2}\right]_{0}^{2} = 10\]Combine to get the result of 55.

Step 6: Final Result

After evaluating all the integrals, the final result of the iterated integral is 55.

Key Concepts

Order of Integration

In calculus, especially when dealing with multiple integrals, it's crucial to understand the order of integration. For iterated integrals, like the one in the original problem, different parts of the integral are evaluated in a specific sequence. Here, the integration must start from the innermost integral and progress outwards. This means that you first integrate with respect to the variable inside the innermost integral, then move to the next variable indicated, and finally, address the outermost variable.
For example, in our original problem, you integrate in the order of \(d z\), \(d y\), and finally \(d x\). Each step needs careful substitution of results from previous steps. This structured order ensures correct evaluation and simplifies complex calculations by breaking them down into smaller, manageable tasks.

Calculus Problem-Solving

Calculus problem-solving often stands as a cornerstone for tackling a wide range of scientific and engineering problems. Here, it's about reducing complexity through strategic division into simpler parts. Starting with the innermost region of a problem—represented by the innermost integral—allows you to solve piece by piece.
- **Break It Down:** Look at what each section of an integral does. Evaluate step by step. - **Simplify:** When tackling the innermost integral, solve for constants or simple expressions. This helps to reduce unnecessary complexity moving forward. - **Substitute Smartly:** Each evaluation provides information for the next integration step. Use these outcomes intelligently to streamline subsequent calculations.

Evaluation of Definite Integrals

The evaluation of definite integrals requires attentiveness to the limits of integration and the behavior of the function within these bounds. As seen in the solution steps, you start by evaluating the innermost integral using its specific bounds and then substitute the result into the subsequent integral.
- **Innermost Integral:** Focus first on the bounds of the innermost integral. Solve, then take this result to use in the next part. - **Middle and Final Steps:** After calculating for the first layer, use its result to inform the next integral. This carefully staged approach assures precision and accuracy. - **Combine Results:** Pull together all solved parts to finalize the evaluation, which often involves simple arithmetic after reaching the outermost integral.

Multiple Integrals

Multiple integrals allow us to compute volume, area, and other properties over a multi-dimensional space. They involve solving integrals that have more than one integration process, often used in physics and engineering. The problem we tackled is a triple integral, meaning there are three levels of integration.- **Understanding Scope:** Knowing what each integral dimension represents helps. For example, \(z\) might represent height, with \(y\) and \(x\) covering length and width.- **Dealing with Complexity:** Use each calculation step to minimize function complexity. Essentially, the more complex the function within the bounds, the more challenging but precise the result.By understanding multiple integrals, we unlock the ability to solve real-world problems involving volume calculation or multi-variable problems in varied fields. Always remember, the key is breaking it down into smaller, digestible problems.