Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate each of the iterated integrals. $$ \int_{0}^{2} \int_{1}^{3} x^{2} y d y d x $$

Step-by-Step Solution

Answer

The value of the integral is \(\frac{32}{3}\).

Step 1: Understand the Iterated Integral

The given problem is an iterated integral where you need to integrate with respect to \(y\) first and then with respect to \(x\). The integrand is \(x^2 y\) with the inner integral \( \int_{1}^{3} x^2 y \, dy \) and the outer integral \( \int_{0}^{2} \, dx \).

Step 2: Perform the Inner Integration

The inner integral is \( \int_{1}^{3} x^2 y \, dy \). Since \(x^2\) is constant with respect to \(y\), treat it as a constant:\[\int_{1}^{3} x^2 y \, dy = x^2 \left[ \frac{y^2}{2} \right]_{1}^{3} = x^2 \left( \frac{3^2}{2} - \frac{1^2}{2} \right)\] Simplify this:\[x^2 \left( \frac{9}{2} - \frac{1}{2} \right) = x^2 \cdot \frac{8}{2} = 4x^2\]

Step 3: Perform the Outer Integration

Now integrate \(4x^2\) with respect to \(x\) from 0 to 2:\[\int_{0}^{2} 4x^2 \, dx = 4 \int_{0}^{2} x^2 \, dx\]First, find the antiderivative:\[\int x^2 \, dx = \frac{x^3}{3}\]Therefore:\[4 \left[ \frac{x^3}{3} \right]_{0}^{2} = 4 \left( \frac{2^3}{3} - \frac{0^3}{3} \right) = 4 \cdot \frac{8}{3}\]This simplifies to:\[\frac{32}{3}\]

Step 4: Conclusion

The evaluation of the iterated integrals results in a final value of \(\frac{32}{3}\). This completes the integration process.

Key Concepts

Multiple Integrals

Multiple integrals involve integrating functions with more than one variable. They are an extension of single-variable integrals used in Calculus. In our exercise, we are dealing with a double integral, which involves two variables: \(x\) and \(y\). The process requires performing integration more than once, iteratively or nested one inside the other.

• Double integrals compute the volume under a surface within a given range for variables.
• The order of integration is essential; here, we integrate with respect to \(y\) first, then \(x\).
• The limits of integration for \(y\) are from 1 to 3 and for \(x\) from 0 to 2.

Understanding multiple integrals is crucial for solving more advanced problems in fields like physics and engineering, where these methods are often used to calculate areas, volumes, and other related properties.

Integration Techniques

Integration techniques are methods or strategies used to evaluate integrals effectively. For multiple integrals, the technique often involves breaking down the problem into manageable steps by first solving one integral and then the other.

• Identify constants: In the inner integral \( \int_{1}^{3} x^2 y \, dy \), we treat \(x^2\) as constant since we integrate with respect to \(y\).
• Find antiderivatives: Compute the antiderivative of the function with respect to the variable of integration.
• Substitute limits: Plug in the upper and lower limits into the antiderivative to find values. Simplify the result.

These standard integration techniques are the building blocks to handling iterated integrals successfully in calculus problems.

Calculus Problem Solving

Calculus problem-solving involves a structured approach to unravel complex mathematical problems using calculus concepts. When tackling iterated integrals:

• Analyze the structure: Determine the order and method of integration.
• Simplify step by step: Break down the problem into smaller parts, solve each part, and combine results.
• Check your answers: Verify the calculations for accuracy, ensuring each step is clearly followed.

Applying these steps develops problem-solving skills critical in advancing mathematical proficiency and application in various scientific and engineering domains.

Definite Integrals

Definite integrals have fixed upper and lower limits, which makes them indispensable for computing exact quantities such as areas and volumes. In the exercise, definite integrals are used at each step:

• They provide exact values for the integration region defined by the limits.
• The evaluated definite integral gives the final value of \( \frac{32}{3} \) for the problem.
• Verification ensures that these computations reflect the actual problem dynamics.

Using definite integrals often requires precision and careful attention to the bounds and the evaluated function, helping in the determination of exact answers in real-world contexts.