Evaluating The Triple Integral Of (x^2 + Ln(y) + Z) Over A Rectangular Box

This article delves into the process of evaluating a triple integral over a rectangular box. Specifically, we will tackle the integral:

${ \iiint_B (x^2 + \ln(y) + z) \, dV }$

where the region B is defined as the rectangular box:

${ B = \{(x, y, z) \mid 0 \leq x \leq 8, 1 \leq y \leq 9, 0 \leq z \leq 2\} }$

We will walk through the step-by-step calculation, explaining the techniques and reasoning behind each step. The final answer will be rounded to four decimal places. Understanding triple integrals is crucial 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 provides a clear illustration of how to evaluate a triple integral over a rectangular region, a fundamental skill in multivariable calculus. The problem at hand requires us to integrate a function of three variables, namely x, y, and z, over a three-dimensional region B. The function to be integrated is the sum of three terms: x squared, the natural logarithm of y, and z. The region B is defined by the inequalities 0 ≤ x ≤ 8, 1 ≤ y ≤ 9, and 0 ≤ z ≤ 2. This represents a rectangular box in three-dimensional space, where x varies from 0 to 8, y varies from 1 to 9, and z varies from 0 to 2. The integral ${\iiint_B (x^2 + \ln(y) + z) \, dV}$ represents the triple integral of the function f(x, y, z) = x² + ln(y) + z over the region B. To evaluate this integral, we will perform a sequence of single integrations, one for each variable. The order of integration can be chosen freely, but it is often convenient to integrate in the order dz dy dx or dx dy dz, depending on the specific problem and the function being integrated. In this case, since the limits of integration are constants for all variables, the order of integration does not affect the complexity of the calculation. We will choose the order dz dy dx. This means we will first integrate with respect to z, then with respect to y, and finally with respect to x.

Step-by-Step Solution

1. Setting up the Iterated Integral

The triple integral can be expressed as an iterated integral:

${ \iiint_B (x^2 + \ln(y) + z) \, dV = \int_0^8 \int_1^9 \int_0^2 (x^2 + \ln(y) + z) \, dz \, dy \, dx }$

This iterated integral represents the same triple integral as the original expression, but it breaks down the integration process into three single integrals. We will evaluate these integrals one at a time, starting with the innermost integral. The key idea here is to treat the other variables as constants while integrating with respect to one variable. This allows us to simplify the integration process and apply the rules of single-variable calculus. The limits of integration for each variable are determined by the inequalities that define the region B. For example, the limits of integration for z are 0 and 2, which correspond to the lower and upper bounds of z in the region B. Similarly, the limits of integration for y are 1 and 9, and the limits of integration for x are 0 and 8. The choice of the order of integration is often a matter of convenience. In this case, since the limits of integration are constants for all variables, the order of integration does not significantly affect the complexity of the calculation. However, in other cases, choosing the right order of integration can greatly simplify the process. For example, if the limits of integration depend on other variables, it may be advantageous to integrate with respect to those variables first. This can help to avoid complicated integrals and make the calculation more manageable. In the next step, we will evaluate the innermost integral, which is the integral with respect to z. We will treat x and y as constants and apply the rules of integration to find the antiderivative of the integrand with respect to z. This will give us a function that depends on x, y, and z, which we will then evaluate at the limits of integration for z. The result will be a function of x and y only, which we will then integrate with respect to y in the next step.

2. Integrating with Respect to z

First, we integrate with respect to z, treating x and y as constants:

${ \int_0^2 (x^2 + \ln(y) + z) \, dz = \left[x^2z + z\ln(y) + \frac{1}{2}z^2\right]_0^2 }$

Evaluating the expression at the limits of integration (2 and 0) gives us:

${ \left(x^2(2) + 2\ln(y) + \frac{1}{2}(2)^2\right) - \left(x^2(0) + 0\ln(y) + \frac{1}{2}(0)^2\right) = 2x^2 + 2\ln(y) + 2 }$

Now we have reduced the triple integral to a double integral:

${ \int_0^8 \int_1^9 (2x^2 + 2\ln(y) + 2) \, dy \, dx }$

This step demonstrates the process of evaluating a definite integral. We first find the antiderivative of the integrand with respect to the variable of integration, which in this case is z. The antiderivative is a function that, when differentiated, gives us the original integrand. We then evaluate the antiderivative at the upper and lower limits of integration and subtract the values. This gives us the value of the definite integral. In this case, the antiderivative of x² + ln(y) + z with respect to z is x²z + zln(y) + (1/2)z². We evaluate this expression at z = 2 and z = 0 and subtract the values to get 2x² + 2ln(y) + 2. This is the result of the integration with respect to z. The next step is to integrate this result with respect to y. We will treat x as a constant and apply the rules of integration to find the antiderivative of the expression with respect to y. This will give us a function that depends on x and y, which we will then evaluate at the limits of integration for y. The result will be a function of x only, which we will then integrate with respect to x in the final step. The process of evaluating a triple integral involves a sequence of single integrations. Each integration reduces the dimensionality of the integral by one. In this case, we started with a triple integral, which is an integral over a three-dimensional region. After the first integration with respect to z, we obtained a double integral, which is an integral over a two-dimensional region. After the second integration with respect to y, we will obtain a single integral, which is an integral over a one-dimensional interval. Finally, the third integration with respect to x will give us a numerical value, which is the value of the original triple integral.

3. Integrating with Respect to y

Next, we integrate with respect to y:

${ \int_1^9 (2x^2 + 2\ln(y) + 2) \, dy = \left[2x^2y + 2(y\ln(y) - y) + 2y\right]_1^9 }$

The antiderivative of ${2\ln(y)}$ is found using integration by parts. Evaluating at the limits:

${ \left(2x^2(9) + 2(9\ln(9) - 9) + 2(9)\right) - \left(2x^2(1) + 2(1\ln(1) - 1) + 2(1)\right) }$

Simplifying, we get:

${ 18x^2 + 18\ln(9) - 18 + 18 - 2x^2 - 2(0 - 1) - 2 = 16x^2 + 18\ln(9) - 2 + 2 = 16x^2 + 18\ln(9) }$

Now we have a single integral:

${ \int_0^8 (16x^2 + 18\ln(9)) \, dx }$

This step highlights the importance of knowing the integrals of common functions, as well as techniques like integration by parts. Integration by parts is a powerful technique for integrating products of functions. It is based on the product rule for differentiation and allows us to rewrite an integral in a form that is easier to evaluate. In this case, we used integration by parts to find the antiderivative of 2ln(y). The formula for integration by parts is ∫ u dv = uv - ∫ v du, where u and v are functions of the variable of integration. The key to using integration by parts is to choose u and dv in such a way that the integral ∫ v du is easier to evaluate than the original integral ∫ u dv. In this case, we chose u = ln(y) and dv = 2 dy. This gives us du = (1/y) dy and v = 2y. Plugging these into the formula for integration by parts, we get ∫ 2ln(y) dy = 2yln(y) - ∫ 2 dy = 2yln(y) - 2y + C, where C is the constant of integration. After finding the antiderivative, we evaluate it at the limits of integration, which are 1 and 9 in this case. We subtract the value of the antiderivative at the lower limit from the value at the upper limit to get the value of the definite integral. This gives us 16x² + 18ln(9). This is the result of the integration with respect to y. The final step is to integrate this result with respect to x. We will apply the power rule for integration to find the antiderivative of 16x² and the constant rule to find the antiderivative of 18ln(9). We will then evaluate the antiderivative at the limits of integration for x and subtract the values to get the final answer.

4. Integrating with Respect to x

Finally, we integrate with respect to x:

${ \int_0^8 (16x^2 + 18\ln(9)) \, dx = \left[\frac{16}{3}x^3 + 18\ln(9)x\right]_0^8 }$

Evaluating at the limits:

${ \left(\frac{16}{3}(8)^3 + 18\ln(9)(8)\right) - \left(\frac{16}{3}(0)^3 + 18\ln(9)(0)\right) = \frac{16}{3}(512) + 144\ln(9) }$

${ = \frac{8192}{3} + 144\ln(9) }$

This final step brings together all the previous calculations to arrive at the final answer. We integrate the resulting expression from the previous step with respect to x. This involves finding the antiderivative of 16x² + 18ln(9) with respect to x. The antiderivative of 16x² is (16/3)x³, and the antiderivative of 18ln(9) is 18ln(9)x. We then evaluate the antiderivative at the upper and lower limits of integration, which are 8 and 0 in this case. We subtract the value of the antiderivative at the lower limit from the value at the upper limit to get the final answer. The final answer is (8192/3) + 144ln(9). This is the exact value of the triple integral. To get a numerical approximation, we can use a calculator to evaluate this expression. The natural logarithm of 9 is approximately 2.19722457734. Plugging this into the expression, we get (8192/3) + 144(2.19722457734) ≈ 2730.66666667 + 316.400349137 ≈ 3047.0670158. Rounding this to four decimal places, we get 3047.0670. This is the final answer to the problem. The process of evaluating a triple integral over a rectangular box involves a sequence of single integrations. The order of integration can be chosen freely, but it is often convenient to integrate in the order dz dy dx or dx dy dz, depending on the specific problem and the function being integrated. The key idea is to treat the other variables as constants while integrating with respect to one variable. This allows us to simplify the integration process and apply the rules of single-variable calculus. The limits of integration for each variable are determined by the inequalities that define the region of integration. The final answer is a numerical value, which represents the value of the triple integral.

5. Numerical Approximation

Now, we approximate the result to four decimal places:

${ \frac{8192}{3} + 144\ln(9) \approx 2730.6667 + 144(2.1972) \approx 2730.6667 + 316.4003 \approx 3047.0670 }$

Final Answer

The value of the triple integral, rounded to four decimal places, is approximately 3047.0670.

Conclusion

We have successfully evaluated the triple integral ${\iiint_B (x^2 + \ln(y) + z) \, dV}$ over the rectangular box B. The process involved breaking down the triple integral into a sequence of single integrals, integrating with respect to each variable in turn, and finally approximating the result to the desired level of accuracy. This problem serves as a great example for understanding the practical application and step-by-step evaluation of a triple integral, which is a fundamental concept in multivariable calculus and has applications in various scientific and engineering fields. The key to evaluating triple integrals is to break them down into a sequence of single integrals. This allows us to apply the rules of single-variable calculus to each integration step. The order of integration can be chosen freely, but it is often convenient to integrate in the order dz dy dx or dx dy dz, depending on the specific problem and the function being integrated. The key idea is to treat the other variables as constants while integrating with respect to one variable. This simplifies the integration process and makes it easier to find the antiderivative. The limits of integration for each variable are determined by the inequalities that define the region of integration. These inequalities specify the range of values that each variable can take within the region. The final answer is a numerical value, which represents the value of the triple integral. This value can be interpreted as the volume of a region in three-dimensional space, or as the mass of a solid object, depending on the context of the problem. Triple integrals are used in various scientific and engineering fields to calculate volumes, masses, centers of mass, moments of inertia, and other properties of three-dimensional objects. They are also used in computer graphics to render three-dimensional scenes and to perform other calculations. Understanding triple integrals is essential for anyone working in these fields. The example presented in this article provides a clear illustration of how to evaluate a triple integral over a rectangular box. By following the steps outlined in this article, you can learn how to solve similar problems and apply the concept of triple integrals to a wide range of applications.