Evaluating The Triple Integral Of Xyz A Step-by-Step Solution
The realm of multivariable calculus unveils the power of integration in higher dimensions. Triple integrals, in particular, extend the concept of definite integrals to functions of three variables over a three-dimensional region. In this article, we will delve into the evaluation of a specific triple integral, providing a step-by-step solution and highlighting the key concepts involved.
The problem at hand is to find the value of the triple integral:
∫01 ∫0x ∫0y xyz dz dy dx
This integral represents the volume-weighted sum of the function f(x, y, z) = xyz over the region defined by the limits of integration. Understanding how to set up and solve such integrals is crucial in various fields, including physics, engineering, and computer graphics.
Before we embark on the integration process, it is essential to visualize the region over which we are integrating. The limits of integration provide us with this information. Let's break them down:
- 0 ≤ x ≤ 1: This tells us that the region is bounded by the planes x = 0 and x = 1 in the x-direction.
- 0 ≤ y ≤ x: This inequality indicates that for a given x, y ranges from 0 to x. This implies that we are within the region bounded by the planes y = 0 and y = x.
- 0 ≤ z ≤ y: Similarly, for given x and y, z varies from 0 to y. This means we are within the region bounded by the planes z = 0 and z = y.
Putting it all together, the region of integration is a tetrahedron in the first octant, bounded by the coordinate planes (x = 0, y = 0, z = 0) and the planes y = x and z = y. Visualizing this region helps in understanding the flow of integration and verifying the correctness of the solution.
Now that we have a clear picture of the region of integration, we can proceed with the evaluation of the triple integral. The key is to integrate iteratively, starting with the innermost integral and working our way outwards. Let's break down the process:
1. Integrate with Respect to z
We begin by integrating the function xyz with respect to z, treating x and y as constants:
∫0y xyz dz = xy [z2/2]0y = xy (y2/2 - 0) = xy3/2
So, the innermost integral evaluates to xy3/2.
2. Integrate with Respect to y
Next, we integrate the result from the previous step, xy3/2, with respect to y, treating x as a constant:
∫0x (xy3/2) dy = x [y4/8]0x = x (x4/8 - 0) = x5/8
Now, we have reduced the integral to a single variable, x.
3. Integrate with Respect to x
Finally, we integrate x5/8 with respect to x over the interval [0, 1]:
∫01 (x5/8) dx = [x6/48]01 = (1/48 - 0) = 1/48
Therefore, the value of the triple integral is 1/48.
To solidify our understanding, let's present the calculation in a more detailed format:
∫01 ∫0x ∫0y xyz dz dy dx
= ∫01 ∫0x [ ∫0y xyz dz ] dy dx
= ∫01 ∫0x [ xy (z2/2) |0y ] dy dx
= ∫01 ∫0x xy (y2/2) dy dx
= ∫01 ∫0x (xy3/2) dy dx
= ∫01 [ ∫0x (xy3/2) dy ] dx
= ∫01 [ x (y4/8) |0x ] dx
= ∫01 x (x4/8) dx
= ∫01 (x5/8) dx
= (1/8) ∫01 x5 dx
= (1/8) [ x6/6 |01 ]
= (1/8) (1/6)
= 1/48
This step-by-step breakdown clarifies the application of the fundamental theorem of calculus in multiple dimensions.
While we solved the integral by integrating in the order dz dy dx, it's important to recognize that the order of integration can sometimes be changed. However, altering the order requires careful consideration of the limits of integration to ensure the region is correctly represented. In some cases, changing the order can simplify the integration process, while in others, it might make it more complex. For this particular problem, the chosen order seemed quite natural and efficient.
Triple integrals are not merely mathematical curiosities; they have significant applications in various scientific and engineering disciplines. Some key applications include:
- Calculating Volumes: Triple integrals can be used to determine the volume of a three-dimensional region. By integrating the constant function f(x, y, z) = 1 over the region, we obtain its volume.
- Finding Mass and Center of Mass: If we have a three-dimensional object with a varying density ρ(x, y, z), we can use triple integrals to calculate its total mass and the coordinates of its center of mass.
- Calculating Moments of Inertia: Moments of inertia are crucial in understanding the rotational behavior of rigid bodies. Triple integrals provide the tool to calculate these moments for objects with complex shapes and mass distributions.
- Fluid Dynamics: Triple integrals play a role in calculating the flux of a fluid through a surface or the total amount of a substance flowing through a region.
- Electromagnetism: In electromagnetism, triple integrals are used to calculate electric and magnetic fields generated by charge and current distributions.
These are just a few examples, and the versatility of triple integrals makes them an indispensable tool in many areas of science and engineering.
When evaluating triple integrals, it's easy to make mistakes, especially when dealing with the limits of integration. Here are some common pitfalls to watch out for:
- Incorrect Order of Integration: As mentioned earlier, the order of integration matters. Changing the order without adjusting the limits accordingly will lead to an incorrect result.
- Incorrect Limits of Integration: The limits must accurately reflect the region of integration. A common mistake is to mix up the variables or to not account for the dependencies between them.
- Algebraic Errors: Like any integration problem, algebraic errors can creep in. It's crucial to double-check each step of the calculation.
- Forgetting the Jacobian: In some coordinate systems (like cylindrical or spherical), a Jacobian factor needs to be included in the integral. Forgetting this factor will lead to an incorrect answer.
- Not Visualizing the Region: As emphasized earlier, visualizing the region of integration is crucial. A clear picture of the region helps in setting up the limits correctly and in spotting potential errors.
By being mindful of these common mistakes, you can significantly improve your accuracy in evaluating triple integrals.
In this article, we have successfully evaluated the triple integral ∫01 ∫0x ∫0y xyz dz dy dx and found its value to be 1/48. We emphasized the importance of understanding the region of integration, the step-by-step process of iterative integration, and the various applications of triple integrals in science and engineering. Furthermore, we highlighted common mistakes to avoid, helping you develop a more robust approach to solving these types of problems. Mastering triple integrals is a valuable skill for anyone working in quantitative fields, and we hope this article has provided you with a clear and comprehensive understanding of the concepts involved. Remember, the key is to practice, visualize, and pay close attention to detail!
