Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

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 Integrals Order

The given integral is a triple integral with the order of integration: \( dz \), \( dy \), \( dx \). It means we will first integrate with respect to \( z \), then \( y \), and finally with respect to \( x \).

Step 2: Evaluate the Inner Integral

Evaluate the innermost integral with respect to \( z \). The integral \( \int_{0}^{3y+x} d z \) with limits from \( 0 \) to \( 3y+x \) results in: \( z \bigg|_0^{3y+x} = (3y+x) - 0 = 3y+x \).

Step 3: Evaluate the Middle Integral

Now integrate \( (3y+x) \) with respect to \( y \) from \( -1 \) to \( 4 \).\[ \int_{-1}^{4} (3y+x) \, dy = \left( \frac{3}{2}y^2 + xy \right) \bigg|_{-1}^{4} \]Calculate:\[ \left( \frac{3}{2}\times 4^2 + 4x \right) - \left( \frac{3}{2}\times (-1)^2 + (-x) \right) = (24 + 4x) - \left( \frac{3}{2} - x \right) = 24 + 4x - 1.5 + x = 22.5 + 5x \]

Step 4: Final Integration with respect to x

Integrate the expression \( 22.5 + 5x \) with respect to \( x \) from \( 0 \) to \( 2 \).\[ \int_{0}^{2} (22.5 + 5x) \, dx = \left( 22.5x + \frac{5}{2}x^2 \right) \bigg|_0^2 \]Calculate:\[ \left( 22.5\times 2 + \frac{5}{2}\times 2^2 \right) - \left( 0 \right) = 45 + 10 = 55 \]

Step 5: Conclusion

The result of the triple integral \( \int_{0}^{2} \int_{-1}^{4} \int_{0}^{3 y+x} dz \, dy \, dx \) is \( 55 \).

Key Concepts

Triple Integral

A triple integral is essentially an extension of the concept of double integrals to three dimensions. It is used to compute the volume under a surface in a three-dimensional space defined by three variables. In this exercise, the triple integral is given by \( \int_{0}^{2} \int_{-1}^{4} \int_{0}^{3 y+x} d z \, dy \, dx \), where we are integrating over the variables \( z \), \( y \), and \( x \). Each integral has upper and lower limits that specify the region of integration. These integrals are significant in finding volumes, mass, or charge distributions when a region is defined in three dimensions.
Triple integrals can sometimes be tricky because of their complexity, but they become more understandable when viewed as a sequence of nested integrals, each performed one after the other from the inside out.

Order of Integration

The order of integration in a multiple integral refers to the sequence in which the integrations are performed. For a triple integral like \( \int_{0}^{2} \int_{-1}^{4} \int_{0}^{3 y+x} d z \, dy \, dx \), the order is \( dz, dy, dx \). This means you first integrate with respect to \( z \), then with respect to \( y \), and finally with respect to \( x \).
Choosing the correct order of integration can simplify the problem, and sometimes, it's necessary to change the order to make the calculation easier or possible. The order is determined based on the given limits of integration and the function being integrated. If the region of integration is complex, analyzing the region carefully is crucial to determine the optimal order of integration.

Integration process

The integration process for multiple integrals is streamlined by working from the innermost integral to the outermost. Let's break down the integration process step by step for the triple integral at hand:

• **Evaluate the innermost integral:** Start with \( \int_{0}^{3y+x} d z \). The variable \( z \) is integrated out, yielding the result \( 3y + x \).
• **Evaluate the middle integral:** Next, integrate \( 3y + x \) with respect to \( y \) from \( -1 \) to \( 4 \). This results in \( \frac{3}{2}y^2 + xy \), evaluated to get \( 22.5 + 5x \).
• **Final integration with respect to \( x \):** Now, the expression \( 22.5 + 5x \) is integrated with respect to \( x \) from \( 0 \) to \( 2 \). After performing this integration, you arrive at the result \( 55 \).

Breaking down the integration process in this way allows you to tackle what could be a daunting task one step at a time, making it more manageable and more intuitive.

Limits of Integration

The limits of integration in an iterated or triple integral play a crucial role. They determine the range over which each variable is integrated. For the problem \( \int_{0}^{2} \int_{-1}^{4} \int_{0}^{3 y+x} dz \, dy \, dx \), the limits are clearly defined as:

• \( z \) ranges from \( 0 \) to \( 3y + x \).
• \( y \) ranges from \( -1 \) to \( 4 \).
• \( x \) ranges from \( 0 \) to \( 2 \).

These limits help define the bounds of the region in three-dimensional space over which the integration occurs. They can be constants, as seen with \( y \) and \( x \), or they can be functions of the other variables, as seen with \( z \) in this problem.
Understanding how these limits translate into regions in space is critical. It allows you to visualize the region of integration, making it easier to solve or manipulate the integral as needed. Analyzing the limits helps to anticipate challenges within the execution of integration, particularly when changing the order of integration is necessary.