Solving Double Integral T = ∫₀⁶ ∫₀⁸ (2xy + 2x - X² - 2ay² + 72) Dx Dy A Step-by-Step Guide

Introduction to Double Integrals

In the realm of multivariable calculus, double integrals play a pivotal role in calculating volumes, areas, and other crucial quantities across two-dimensional regions. Double integrals extend the concept of single integrals from functions of one variable to functions of two variables. This article aims to provide a detailed, step-by-step guide on how to evaluate the double integral T = ∫₀⁶ ∫₀⁸ (2xy + 2x - x² - 2ay² + 72) dx dy. Understanding the process of evaluating double integrals is essential for students, engineers, and anyone working in fields that require advanced mathematical techniques. This integral not only serves as a practical example but also illuminates the broader applications of double integrals in diverse scientific and engineering contexts.

To truly appreciate the utility of double integrals, it's important to grasp the fundamental concept. Imagine you have a surface defined by a function z = f(x, y) over a region in the xy-plane. The double integral of f(x, y) over this region gives you the volume between the surface and the xy-plane. This has direct applications in fields such as physics, where you might calculate the mass of a two-dimensional object with varying density, or in engineering, where you might determine the center of mass of a plate. In economics, double integrals can even be used to calculate consumer surplus over a region of the market. Therefore, mastering double integrals is not just an academic exercise; it’s a powerful tool for solving real-world problems. The ability to correctly set up and evaluate these integrals is a cornerstone of advanced mathematical proficiency, allowing for accurate modeling and analysis in a wide range of disciplines. This article will break down the process into manageable steps, ensuring that even those new to the concept can follow along and gain a solid understanding.

Breaking Down the Integral T = ∫₀⁶ ∫₀⁸ (2xy + 2x - x² - 2ay² + 72) dx dy

The given double integral is T = ∫₀⁶ ∫₀⁸ (2xy + 2x - x² - 2ay² + 72) dx dy. This mathematical expression represents the integral of the function f(x, y) = 2xy + 2x - x² - 2ay² + 72 over a rectangular region in the xy-plane. The limits of integration, 0 to 8 for x and 0 to 6 for y, define the boundaries of this region. Evaluating this integral involves a sequential process, where we first integrate with respect to one variable (either x or y) while treating the other variable as a constant, and then integrate the result with respect to the remaining variable. This stepwise approach is crucial for managing the complexity of multivariable calculus. The function itself, 2xy + 2x - x² - 2ay² + 72, is a polynomial in two variables, making it a relatively straightforward function to integrate. However, the presence of the parameter 'a' adds a layer of complexity, as the final result will be an expression in terms of 'a'.

The key to successfully evaluating this double integral lies in understanding the order of integration and carefully applying the rules of calculus. We begin by focusing on the inner integral, which is with respect to x. This means we treat y as a constant and integrate the function 2xy + 2x - x² - 2ay² + 72 with respect to x. Once we've evaluated this inner integral, we obtain a function that depends only on y. We then integrate this resulting function with respect to y, using the limits of integration for y. This sequential process transforms the double integral into a series of single-variable integrals, which are much easier to handle. By meticulously following these steps, we can systematically break down the problem and arrive at the correct solution. The presence of 'a' simply means we'll carry it through the calculations as a constant, and the final result will reflect its influence on the integral's value. This method of breaking down a complex problem into smaller, manageable parts is a fundamental strategy in calculus and is applicable to a wide range of mathematical challenges.

Step-by-Step Evaluation of the Double Integral

The process of evaluating the double integral T = ∫₀⁶ ∫₀⁸ (2xy + 2x - x² - 2ay² + 72) dx dy involves several key steps. First, we tackle the inner integral, treating y as a constant. This involves integrating the function 2xy + 2x - x² - 2ay² + 72 with respect to x, from the limits 0 to 8. Careful application of the power rule and the constant multiple rule of integration is crucial here. Each term in the function is integrated separately, ensuring no algebraic errors are made. For instance, the term 2xy integrates to x²y, 2x integrates to x², x² integrates to (1/3)x³, and so on. The constant 72 integrates to 72x. The term -2ay² is treated as a constant multiple, so it integrates to -2ay²x. Once we've integrated each term, we evaluate the result at the limits of integration, 8 and 0, and subtract the values to find the definite integral.

The next step is to integrate the result from the inner integral with respect to y. The expression obtained after evaluating the inner integral will be a function of y, possibly involving the parameter 'a'. This function is then integrated with respect to y, from the limits 0 to 6. Again, the power rule and the constant multiple rule are applied meticulously. For example, terms like y² will integrate to (1/3)y³, and constant terms will integrate to linear terms in y. The critical aspect here is to handle the 'a' term correctly, treating it as a constant. Once the integration is complete, we evaluate the result at the limits of integration, 6 and 0, and subtract the values to obtain the final result. This final value is the definite value of the double integral, and it will likely be an expression in terms of 'a'. By breaking the double integral into these manageable steps – integrating with respect to x first, then with respect to y – we can systematically solve the problem and arrive at the correct answer. Each step requires careful attention to detail and a solid understanding of basic integration techniques.

Performing the Inner Integral with Respect to x

The first crucial step in solving T = ∫₀⁶ ∫₀⁸ (2xy + 2x - x² - 2ay² + 72) dx dy is to evaluate the inner integral. This means we integrate the function 2xy + 2x - x² - 2ay² + 72 with respect to x, treating y and a as constants. This process is fundamental to understanding how double integrals work, as it simplifies a two-dimensional problem into a series of one-dimensional problems. To begin, we apply the basic rules of integration term by term. The integral of 2xy with respect to x is x²y, because we increase the power of x by one (from 1 to 2) and divide by the new power. Similarly, the integral of 2x is x², the integral of -x² is -(1/3)x³, and the integral of -2ay² (a constant with respect to x) is -2ay²x. The constant term 72 integrates to 72x. Combining these results, we get the antiderivative:

∫ (2xy + 2x - x² - 2ay² + 72) dx = x²y + x² - (1/3)x³ - 2ay²x + 72x

Next, we evaluate this antiderivative at the limits of integration for x, which are 0 and 8. This means we substitute x = 8 and x = 0 into the antiderivative and subtract the results. Substituting x = 8, we get:

(8²)(y) + 8² - (1/3)(8³) - 2a(y²)(8) + 72(8) = 64y + 64 - (512/3) - 16ay² + 576

Substituting x = 0, all terms become zero, so the result is simply 0. Therefore, the definite integral with respect to x is:

64y + 64 - (512/3) - 16ay² + 576 - 0 = 64y - 16ay² + 64 + 576 - (512/3) = 64y - 16ay² + 640 - (512/3)

This simplifies to:

64y - 16ay² + (1920/3) - (512/3) = 64y - 16ay² + (1408/3)

This expression, 64y - 16ay² + (1408/3), is the result of the inner integral and is a function of y. We will use this result as the integrand for the outer integral, which will be integrated with respect to y. This step-by-step process of evaluating the inner integral is crucial for breaking down the double integral into a manageable form, allowing us to proceed with the outer integral and ultimately find the value of T. By carefully applying the rules of integration and evaluating the antiderivative at the limits, we have successfully transformed the inner integral into a simpler expression that depends only on y.

Evaluating the Outer Integral with Respect to y

Following the successful evaluation of the inner integral, the next step in solving T = ∫₀⁶ ∫₀⁸ (2xy + 2x - x² - 2ay² + 72) dx dy is to evaluate the outer integral with respect to y. This involves integrating the result obtained from the inner integral, which is 64y - 16ay² + (1408/3), with respect to y, over the limits 0 to 6. This stage is crucial as it completes the process of evaluating the double integral, providing a final numerical answer or an expression in terms of 'a'. The integral of 64y with respect to y is 32y², as we increase the power of y by one (from 1 to 2) and divide by the new power. The integral of -16ay² with respect to y is -(16/3)ay³, following the same rule. The integral of the constant term (1408/3) is (1408/3)y. Therefore, the antiderivative of the function with respect to y is:

∫ (64y - 16ay² + (1408/3)) dy = 32y² - (16/3)ay³ + (1408/3)y

Next, we evaluate this antiderivative at the limits of integration for y, which are 0 and 6. This involves substituting y = 6 and y = 0 into the antiderivative and subtracting the results. Substituting y = 6, we get:

32(6²) - (16/3)a(6³) + (1408/3)(6) = 32(36) - (16/3)a(216) + (1408/3)(6) = 1152 - 1152a + 2816

Substituting y = 0, all terms become zero, so the result is simply 0. Therefore, the definite integral with respect to y is:

1152 - 1152a + 2816 - 0 = 3968 - 1152a

Thus, the final result of the double integral T is 3968 - 1152a. This expression represents the volume (or signed volume) under the surface defined by the original function 2xy + 2x - x² - 2ay² + 72 over the rectangular region defined by the limits of integration. By carefully evaluating both the inner and outer integrals, we have successfully computed the value of the double integral. The final result is a function of 'a', which means that the value of the integral depends on the specific value of 'a'. This step-by-step evaluation highlights the importance of meticulous calculation and a strong understanding of the fundamental rules of calculus.

Final Result and Interpretation

After meticulously performing both the inner and outer integrations, we arrive at the final result for the double integral T = ∫₀⁶ ∫₀⁸ (2xy + 2x - x² - 2ay² + 72) dx dy, which is T = 3968 - 1152a. This expression is a function of the parameter 'a', indicating that the value of the integral varies depending on the specific value of 'a'. The result T = 3968 - 1152a represents the volume under the surface defined by the function f(x, y) = 2xy + 2x - x² - 2ay² + 72 over the rectangular region in the xy-plane bounded by 0 ≤ x ≤ 8 and 0 ≤ y ≤ 6. The interpretation of this volume depends on the context of the problem. In some cases, it may represent a physical volume, while in others, it might represent a generalized volume or a cumulative quantity in a different domain, such as economics or probability.

The significance of the parameter 'a' in the final result cannot be overstated. The term -1152a shows that as 'a' increases, the value of the integral decreases, and vice versa. This suggests that 'a' has a direct influence on the volume under the surface. For instance, if 'a' is a positive value, it reduces the overall volume, whereas if 'a' is negative, it increases the volume. This behavior is crucial for understanding the characteristics of the function f(x, y) and the geometric properties of the surface it represents. Further analysis could involve determining specific values of 'a' that make the integral zero, positive, or negative, which would provide deeper insights into the function's behavior and its applications. In practical terms, the ability to evaluate and interpret double integrals like this is essential in fields such as engineering, physics, and computer graphics, where volume and area calculations are fundamental to problem-solving and design. The final result, T = 3968 - 1152a, is not just a numerical answer; it's a piece of information that provides valuable insights into the underlying mathematical model.

Common Mistakes to Avoid

When evaluating double integrals like T = ∫₀⁶ ∫₀⁸ (2xy + 2x - x² - 2ay² + 72) dx dy, several common mistakes can lead to incorrect results. Being aware of these pitfalls is crucial for achieving accuracy and a solid understanding of the process. One of the most frequent errors is incorrectly applying the order of integration. Double integrals are evaluated sequentially, first with respect to one variable and then the other. It is essential to perform the inner integral first, treating the other variable as a constant, before moving on to the outer integral. Reversing this order or not keeping track of which variable is being integrated can lead to significant errors. Another common mistake is making errors in the integration process itself. This can involve misapplying the power rule, the constant multiple rule, or other fundamental integration techniques. For instance, forgetting to include the constant of integration (although it cancels out in definite integrals) or incorrectly handling negative signs can lead to incorrect antiderivatives and, consequently, incorrect final results. Careful attention to detail and a thorough understanding of integration rules are essential here.

Another source of error lies in the evaluation of the antiderivative at the limits of integration. This step involves substituting the upper and lower limits into the antiderivative and subtracting the results. A common mistake is to forget to subtract or to substitute the limits in the wrong order. It is crucial to evaluate the antiderivative at the upper limit first and then subtract the value obtained at the lower limit. Additionally, algebraic errors in simplifying the expressions after integration and substitution can also lead to incorrect answers. This can involve mishandling fractions, exponents, or combining like terms incorrectly. A systematic approach, where each step is checked carefully, can help minimize these errors. Finally, when dealing with parameters like 'a' in the given integral, it is important to treat them as constants during integration and to carry them through the calculations correctly. Misinterpreting 'a' as a variable or making mistakes in algebraic manipulations involving 'a' can lead to an incorrect final result. By being mindful of these common mistakes and adopting a methodical approach, one can significantly improve their accuracy in evaluating double integrals.

Conclusion: Mastering Double Integrals

In conclusion, the evaluation of the double integral T = ∫₀⁶ ∫₀⁸ (2xy + 2x - x² - 2ay² + 72) dx dy provides a comprehensive illustration of the techniques and considerations involved in multivariable calculus. By systematically breaking down the integral into manageable steps, starting with the inner integral with respect to x and then proceeding to the outer integral with respect to y, we were able to arrive at the final result: T = 3968 - 1152a. This result not only gives us a numerical value (dependent on the parameter 'a') but also offers valuable insights into the behavior of the function and the volume it represents over the specified region. The process highlighted the importance of meticulous application of integration rules, careful evaluation at the limits of integration, and the correct handling of parameters.

Mastering double integrals is a critical skill for anyone working in fields that require advanced mathematical analysis. Whether it's calculating volumes in engineering, determining probabilities in statistics, or modeling complex systems in physics, double integrals provide a powerful tool for solving real-world problems. The step-by-step approach outlined in this article can serve as a guide for tackling other double integrals, even those with more complex functions and limits of integration. Furthermore, understanding the common mistakes to avoid, such as errors in the order of integration, incorrect application of integration rules, or algebraic errors, is essential for achieving accurate results. By practicing and applying these techniques, one can develop confidence and proficiency in evaluating double integrals, opening up a wide range of opportunities in various scientific and engineering disciplines. The journey through this integral not only reinforces mathematical skills but also deepens the appreciation for the elegance and utility of calculus in solving practical problems.