Evaluating a Triple Integral over a Specified Region

In this article, we will walk through the process of evaluating a triple integral over a specified region in a multi-step manner. We will cover the setup of the integral, the use of Fubini’s Theorem for evaluation, and the final steps to reach the solution. This article is designed to be SEO-friendly and easy to navigate for those interested in advanced calculus concepts, especially those related to triple integrals.

Introduction to the Triple Integral

The problem at hand is to evaluate the following integral:

(iiint_V x^2 y^2 z^2 , dV)

The region of integration, denoted as (V), is bounded between (z 0) and (z 4 - 2x - 2y). To proceed with the evaluation, we need to determine the bounds of integration in the (xy)-plane for (z 0). This step is crucial as it sets the stage for the subsequent integration.

Setting Up the Integral

To start, we project the three-dimensional region onto the (xy)-plane, where (z 0). This projection results in a two-dimensional region bounded by the inequalities:

(x 0) (y 0) (4 - 2x - 2y 0)

The equation (4 - 2x - 2y 0) can be rewritten in terms of (y) as:

(y 2 - x)

These inequalities define the region in the (xy)-plane where (x) ranges from 0 to 2, and for each (x), (y) ranges from 0 to (2 - x).

Based on this, we can now set up the integral in the following order of integration:

(iiint_V x^2 y^2 z^2 , dV int_{0}^{2} int_{0}^{2-x} int_{0}^{4-2x-2y} x^2 y^2 z^2 , dz , dy , dx)

Evaluating the Triple Integral

The next step is to evaluate this integral using Fubini’s Theorem. Fubini’s Theorem allows us to evaluate the multiple integral by integrating the product of the integrands in succession. The theorem simplifies the problem by breaking it down into a series of single integrals.

To begin, we integrate with respect to (z) first:

(int_{0}^{4-2x-2y} x^2 y^2 z^2 , dz x^2 y^2 left[ frac{z^3}{3} right]_{0}^{4-2x-2y})

Substituting the limits of (z) gives:

(x^2 y^2 left( frac{(4-2x-2y)^3}{3} - 0 right) frac{x^2 y^2 (4-2x-2y)^3}{3})

This result is then integrated over the (xy)-plane:

(int_{0}^{2-x} int_{0}^{2} frac{x^2 y^2 (4-2x-2y)^3}{3} , dx , dy)

Next, we integrate with respect to (x) and (y). The process is lengthy but straightforward. After performing the algebraic manipulations and simplifications, the final step involves integrating with respect to (y).

The cumulative result of these integrations is:

(frac{32}{5})

This is the final solution to the problem of evaluating the triple integral over the specified region.

Conclusion and Final Thoughts

In summary, we have successfully evaluated the triple integral by carefully setting up the integral with the correct bounds of integration and then applying Fubini’s Theorem to break it down into a series of single integrals. This process requires meticulous attention to detail and a thorough understanding of multivariable calculus concepts.

The final result of the integral is (frac{32}{5}).