Evaluating The Triple Integral ∫1e ∫1log 4 ∫1z Dy Dx Dz

This article provides a step-by-step evaluation of the triple integral ${ \int_1^e \int_1^{\log 4} \int_1^z dy dx dz }$. Triple integrals are a fundamental concept in multivariable calculus, often used to compute volumes, masses, and other quantities in three-dimensional space. The key to solving these integrals is to work from the inside out, evaluating each integral one at a time while treating the other variables as constants. This article aims to provide a clear and detailed explanation of the process, making it accessible to students and enthusiasts alike. We will break down each step, ensuring that the reader understands the logic behind every calculation. Understanding the mechanics of triple integration not only enhances problem-solving skills but also provides a deeper appreciation for the power of calculus in modeling real-world phenomena.

The order of integration is crucial in triple integrals, and in this case, we will integrate with respect to ${ y }$ first, then ${ x }$, and finally ${ z }$. This order is dictated by the limits provided in the integral. By following this systematic approach, we can simplify the complex integral into a series of manageable steps. Furthermore, this example serves as a practical application of the fundamental theorem of calculus, extending it to multiple dimensions. Mastery of these techniques is essential for anyone studying advanced calculus or related fields such as physics and engineering.

Throughout this article, we will emphasize the importance of precision and attention to detail. A small mistake in any step can lead to a significantly different final answer. Therefore, we will meticulously check each calculation and provide clear explanations for every transition. This methodical approach will not only help in solving this particular problem but also instill a valuable habit for tackling other mathematical challenges. By the end of this article, readers should have a solid understanding of how to evaluate triple integrals and be confident in their ability to apply this knowledge to various problems. This skill is invaluable in many scientific and engineering disciplines, where multi-dimensional calculations are common.

Step-by-Step Evaluation

Step 1: Evaluate the Innermost Integral

The innermost integral is ${ \int_1^z dy }$. We integrate with respect to ${ y }$, treating ${ z }$ as a constant. The integral of 1 with respect to ${ y }$ is simply ${ y }$. So we have:

${ \int_1^z dy = [y]_1^z = z - 1 }$

This step is relatively straightforward, but it's crucial to get the limits of integration correct. We substitute the upper limit ${ z }$ and the lower limit 1 into the expression ${ y }$ and subtract the results. The resulting expression ${ z - 1 }$ will then be used in the next integral. Understanding this process is fundamental to successfully evaluating triple integrals. This initial step simplifies the problem by reducing the dimensionality of the integral, making it more manageable for subsequent calculations. Attention to detail in this step ensures that the subsequent calculations are based on a correct foundation. This process illustrates the power of breaking down a complex problem into smaller, more manageable parts, a strategy that is useful in many areas of mathematics and problem-solving.

The fundamental theorem of calculus is at play here, allowing us to evaluate the definite integral by finding the antiderivative of the integrand and then evaluating it at the limits of integration. This theorem is a cornerstone of calculus, and its application here demonstrates its versatility. The careful application of this theorem is critical for accurate evaluation of integrals. Additionally, this step highlights the importance of understanding the relationship between integration and differentiation. Integration is, in essence, the reverse process of differentiation, and recognizing this relationship is essential for mastering calculus. This initial step is not only a calculation but also a reinforcement of core calculus principles.

Step 2: Evaluate the Second Integral

Now we substitute the result from the first integral into the second integral:

${ \int_1^{\log 4} (z - 1) dx }$

Here, we integrate with respect to ${ x }$, and since ${ (z - 1) }$ does not contain ${ x }$, it is treated as a constant. The integral of a constant with respect to ${ x }$ is the constant times ${ x }$. Thus, we have:

${ \int_1^{\log 4} (z - 1) dx = (z - 1) \int_1^{\log 4} dx = (z - 1) [x]_1^{\log 4} = (z - 1) (\log 4 - 1) }$

This step involves another application of the fundamental theorem of calculus. We found the antiderivative with respect to ${ x }$ and evaluated it at the limits of integration. The result is an expression that depends only on ${ z }$, which is essential for the next integration step. The ability to treat ${ z - 1 }$ as a constant in this context is crucial. This highlights the importance of recognizing which variables are relevant in each step of the integration process. By focusing on the variable of integration, we can simplify the calculation and avoid unnecessary complexity. This step demonstrates the strategic simplification that is characteristic of many calculus problems.

Furthermore, this step reinforces the concept of linearity of integration. The constant factor ${ (z - 1) }$ can be pulled out of the integral, which simplifies the calculation significantly. This property of integrals is a powerful tool that can make complex problems more manageable. The careful application of this property is another key to successful integration. This step also shows how the result of one integration becomes the input for the next, creating a chain of calculations that leads to the final answer. This sequential approach is a hallmark of solving multi-step mathematical problems, and understanding it is critical for success in calculus.

Step 3: Evaluate the Outermost Integral

Finally, we substitute the result from the second integral into the outermost integral:

${ \int_1^e (z - 1) (\log 4 - 1) dz }$

We integrate with respect to ${ z }$. The factor ${ (\log 4 - 1) }$ is a constant, so we can pull it out of the integral:

${ (\log 4 - 1) \int_1^e (z - 1) dz }$

Now, we integrate ${ (z - 1) }$ with respect to ${ z }$:

${ \int (z - 1) dz = \frac{1}{2}z^2 - z + C }$

So the definite integral is:

${ (\log 4 - 1) \left[ \frac{1}{2}z^2 - z \right]_1^e }$

Now, we evaluate at the limits of integration:

${ (\log 4 - 1) \left( \left( \frac{1}{2}e^2 - e \right) - \left( \frac{1}{2}(1)^2 - 1 \right) \right) }$

${ (\log 4 - 1) \left( \frac{1}{2}e^2 - e + \frac{1}{2} \right) }$

This final step brings together all the previous calculations. The integration with respect to ${ z }$ is crucial, and the correct application of the limits of integration is essential for obtaining the correct final result. This step demonstrates the cumulative nature of the problem, where each previous step builds upon the others. The expression ${ (\log 4 - 1) }$ is a constant factor that remains throughout the calculation, highlighting the importance of careful handling of constants in integration. The evaluation at the limits ${ 1 }$ and ${ e }$ requires precise substitution and algebraic manipulation. This is a good example of how calculus problems often involve a combination of integration techniques and algebraic skills.

Furthermore, this step reinforces the concept of definite integrals, where the result is a specific numerical value rather than a function. This numerical value represents the quantity being calculated by the integral, which in this case is related to the volume in three-dimensional space. The final answer provides a concrete result that can be interpreted and used in various applications. This process also illustrates the importance of simplifying the expression as much as possible before the final evaluation. By factoring out the constant term and carefully handling the algebraic manipulations, we arrive at a simplified final answer that is easier to interpret and use.

Final Result

The final result of the triple integral is:

${ (\log 4 - 1) \left( \frac{1}{2}e^2 - e + \frac{1}{2} \right) }$

This expression can be further simplified or approximated numerically if desired. The final result encapsulates the entire process of triple integration, demonstrating the power of calculus to solve complex problems. This answer represents a specific value, which could correspond to a volume, mass, or another quantity depending on the context of the problem. Understanding the meaning of this final result is just as important as knowing how to calculate it. This involves interpreting the result within the original problem statement and understanding the units and dimensions involved. The result is a culmination of all the steps, demonstrating the importance of accuracy and attention to detail throughout the integration process.

Conclusion

In this article, we have provided a detailed step-by-step evaluation of the triple integral ${ \int_1^e \int_1^{\log 4} \int_1^z dy dx dz }$. We have shown how to work through the integral from the inside out, carefully evaluating each integral and applying the limits of integration. This process highlights the fundamental techniques of multivariable calculus and the importance of precision in calculations. The final result, ${ (\log 4 - 1) \left( \frac{1}{2}e^2 - e + \frac{1}{2} \right) }$, represents the volume or another quantity in three-dimensional space, depending on the context of the problem. Mastery of these techniques is crucial for students and professionals in mathematics, physics, engineering, and other related fields.

Throughout the article, we have emphasized the importance of understanding the underlying concepts and principles. The fundamental theorem of calculus plays a central role in evaluating definite integrals, and the linearity of integration is a powerful tool for simplifying calculations. The ability to treat certain variables as constants while integrating with respect to another is also crucial for success in multivariable calculus. By carefully applying these principles and techniques, we can confidently tackle a wide range of triple integral problems. This skill is invaluable in many areas of science and engineering, where multi-dimensional calculations are common and necessary for problem-solving.

Furthermore, this example serves as a valuable exercise in problem-solving strategy. Breaking down a complex problem into smaller, more manageable steps is a key skill in mathematics and many other fields. By systematically working through each step of the integration process, we can arrive at the final solution without getting overwhelmed. This methodical approach can be applied to a variety of mathematical challenges, making it a valuable tool for students and professionals alike. Ultimately, the ability to evaluate triple integrals is not just about performing calculations; it is about understanding the underlying concepts and developing a strategic approach to problem-solving.