Solving The Triple Integral ∫₀¹ ∫₀ˣ ∫₀ʸ Xyz Dz Dy Dx A Step-by-Step Guide

Introduction to Triple Integrals

In the realm of multivariable calculus, triple integrals extend the concept of integration to three-dimensional space. They provide a powerful tool for calculating volumes, masses, and other properties of three-dimensional objects. Understanding triple integrals is crucial for various applications in physics, engineering, and other scientific disciplines. The evaluation of triple integrals often involves a systematic approach, where we integrate with respect to one variable at a time, treating the other variables as constants. This process, known as iterated integration, allows us to break down a complex three-dimensional problem into a series of simpler one-dimensional integrations. The order of integration can significantly impact the complexity of the problem, and careful consideration must be given to the limits of integration for each variable. The given problem, ∫₀¹ ∫₀ˣ ∫₀ʸ xyz dz dy dx, presents a classic example of a triple integral that requires a step-by-step solution to arrive at the final answer. This article aims to dissect the problem, providing a comprehensive explanation of each step involved in the evaluation process, making it accessible to students and enthusiasts alike. Understanding the underlying principles and techniques will empower readers to tackle similar problems with confidence and clarity. Furthermore, we will explore the significance of triple integrals in real-world applications, highlighting their importance in various fields. This exploration will provide a broader context for the mathematical concepts discussed and demonstrate their practical relevance.

Problem Statement: ∫₀¹ ∫₀ˣ ∫₀ʸ xyz dz dy dx

The problem at hand involves evaluating the triple integral ∫₀¹ ∫₀ˣ ∫₀ʸ xyz dz dy dx. This notation represents a sequential integration process over a three-dimensional region. The integrand, xyz, is a function of three variables, and the limits of integration define the region over which we are integrating. The innermost integral, ∫₀ʸ xyz dz, indicates that we first integrate with respect to z, treating x and y as constants. The limits of integration for z are 0 and y, which means we are integrating over the range of z values from 0 to y. Once we evaluate this inner integral, we obtain a function of x and y, which then becomes the integrand for the next integral. The middle integral, ∫₀ˣ (result from inner integral) dy, instructs us to integrate with respect to y, treating x as a constant. The limits of integration for y are 0 and x, indicating that we integrate over the range of y values from 0 to x. Finally, the outermost integral, ∫₀¹ (result from middle integral) dx, tells us to integrate with respect to x, with the limits of integration being 0 and 1. This means we integrate over the range of x values from 0 to 1. The order of integration is crucial, and changing the order can significantly alter the complexity of the problem. In this case, we are integrating in the order dz dy dx. The key to solving this triple integral lies in carefully executing each step of the iterated integration process, paying close attention to the limits of integration and the resulting functions at each stage. By systematically working through each integral, we can arrive at the final numerical value of the triple integral. This value represents the result of integrating the function xyz over the specified three-dimensional region.

Step-by-Step Solution

1. Integrate with Respect to z

We begin by tackling the innermost integral: ∫₀ʸ xyz dz. In this step, we treat x and y as constants, focusing solely on integrating the expression with respect to z. The integral of xyz with respect to z is (1/2)xyz², evaluated from z = 0 to z = y. Substituting the limits of integration, we get (1/2)xy(y²) - (1/2)xy(0²) = (1/2)xy³. This result represents the value of the innermost integral and becomes the integrand for the next step. The process of integrating with respect to z effectively collapses the three-dimensional integral into a two-dimensional one, leaving us with a function of x and y. The key to this step is recognizing that x and y are constants in the context of the z integration, allowing us to treat them as coefficients. The power rule of integration, which states that the integral of z^n is (1/(n+1))z^(n+1), is applied here. Understanding this basic rule is essential for successfully evaluating integrals of this type. The result, (1/2)xy³, is a crucial intermediate step, as it forms the basis for the subsequent integrations. A clear understanding of this step is vital for comprehending the overall solution process.

2. Integrate with Respect to y

Moving on, we now integrate the result from the previous step, (1/2)xy³, with respect to y. This gives us the integral ∫₀ˣ (1/2)xy³ dy. Here, we treat x as a constant and integrate the expression with respect to y. The integral of (1/2)xy³ with respect to y is (1/2)x * (1/4)y⁴ = (1/8)xy⁴, evaluated from y = 0 to y = x. Substituting the limits of integration, we get (1/8)x(x⁴) - (1/8)x(0⁴) = (1/8)x⁵. This result represents the value of the second integral and becomes the integrand for the final step. Similar to the previous step, the key here is recognizing that x is a constant in the context of the y integration. The power rule of integration is again applied, this time to the y³ term. The result, (1/8)x⁵, is another crucial intermediate step, as it simplifies the problem further, reducing it to a single integral with respect to x. This step demonstrates the power of iterated integration, where we progressively reduce the dimensionality of the integral by integrating with respect to one variable at a time. The careful application of the limits of integration is essential to ensure the accuracy of the result.

3. Integrate with Respect to x

Finally, we integrate the result from the previous step, (1/8)x⁵, with respect to x. This gives us the integral ∫₀¹ (1/8)x⁵ dx. Integrating (1/8)x⁵ with respect to x yields (1/8) * (1/6)x⁶ = (1/48)x⁶, evaluated from x = 0 to x = 1. Substituting the limits of integration, we get (1/48)(1⁶) - (1/48)(0⁶) = 1/48. Therefore, the value of the triple integral ∫₀¹ ∫₀ˣ ∫₀ʸ xyz dz dy dx is 1/48. This final step completes the evaluation of the triple integral. The power rule of integration is applied one last time, and the limits of integration are carefully substituted to obtain the final numerical result. The value 1/48 represents the culmination of all the previous steps and provides the answer to the given problem. This result demonstrates the effectiveness of the iterated integration technique in solving triple integrals. The systematic approach of integrating with respect to each variable one at a time, while treating the others as constants, allows us to break down a complex problem into manageable steps. The final answer, 1/48, represents the volume under the surface defined by the integrand xyz over the specified region of integration. This value has significance in various applications, depending on the context of the problem.

Final Answer: 1/48

In conclusion, the value of the triple integral ∫₀¹ ∫₀ˣ ∫₀ʸ xyz dz dy dx is 1/48. This result was obtained by systematically evaluating the integral through iterated integration, starting with the innermost integral and working outwards. Each step involved integrating with respect to one variable while treating the others as constants, a crucial technique in solving multivariable integrals. The power rule of integration was applied repeatedly, and the limits of integration were carefully substituted at each stage to ensure accuracy. The final answer, 1/48, represents the definite value of the triple integral over the specified region. This process highlights the importance of a structured approach when dealing with complex mathematical problems. By breaking down the problem into smaller, more manageable steps, we can effectively arrive at the solution. The understanding of triple integrals and their evaluation is essential in various fields, including physics, engineering, and computer graphics, where they are used to calculate volumes, masses, and other properties of three-dimensional objects. This example serves as a valuable illustration of the techniques involved in solving triple integrals and provides a solid foundation for tackling more complex problems in multivariable calculus.

Significance and Applications of Triple Integrals

Triple integrals are not merely abstract mathematical concepts; they hold significant practical value in various scientific and engineering disciplines. Their primary application lies in calculating volumes of three-dimensional regions. Imagine a complex shape in space; a triple integral can precisely determine the amount of space it occupies. This is crucial in fields like architecture and manufacturing, where accurate volume calculations are essential for design and construction. Beyond volume, triple integrals are instrumental in finding the mass of an object with varying density. If the density of an object changes from point to point, a triple integral can accurately compute its total mass. This is particularly relevant in materials science and engineering, where understanding the mass distribution of a component is vital for structural analysis. The center of mass, a critical concept in physics and engineering mechanics, is another quantity that can be determined using triple integrals. The center of mass represents the average position of all the mass in an object, and its calculation is crucial for understanding the object's stability and motion. Furthermore, triple integrals extend to calculating moments of inertia, which quantify an object's resistance to rotational motion. This is essential in designing rotating machinery and ensuring their stable operation. In fluid dynamics, triple integrals are used to calculate the flow rate of fluids through a given region. This has applications in designing pipelines, analyzing weather patterns, and understanding ocean currents. The applications of triple integrals also extend to computer graphics, where they are used for rendering realistic three-dimensional images. By integrating over the volume of an object, computer algorithms can accurately simulate lighting and shading effects. In essence, triple integrals provide a powerful tool for solving a wide range of problems in science and engineering, demonstrating their practical relevance and importance.