91Ó°ÊÓ

To perform a change of variables, especially in the context of triple integration, we need to understand the role of the Jacobian. The Jacobian serves as a bridge between different coordinate systems, ensuring that the volume elements transform accurately. When transforming variables in a triple integral, the Jacobian is a determinant derived from the partial derivatives of the new variables with respect to the old ones. In this exercise, the transformation is given by - \(u = x\)- \(v = xy\)- \(w = 3z\)The Jacobian matrix for this transformation is \[ J = \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z} \ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \ \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} & \frac{\partial w}{\partial z} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ y & x & 0 \ 0 & 0 & 3 \end{bmatrix} \]Calculating the determinant of this Jacobian matrix, we find it to be:\[ |J| = 3x \]This Jacobian accounts for the non-linear stretching of space caused by the transformation, which is crucial when calculating the transformed integral.

Variable transformation is a powerful technique in calculus that simplifies the process of integration by changing the integration variables. In the context of this exercise, we're converting variables from - \((x, y, z)\) space to - \((u, v, w)\) space.The transformation equations - \(u = x\),- \(v = xy\), and- \(w = 3z\)are chosen based on simplicity and suitability for the given problem.These transformations condense the originally complex integrand into a simpler form, making it easier to evaluate. It simplifies regions of integration and can turn complex shapes into cubes or parallelepipeds which are easier to handle mathematically. Moreover, by using this method in the calculation, we apply the Jacobian to adjust the integral according to the new volume elements.

Changing variables is essentially about introducing a new frame of reference that makes solving a problem more straightforward. In integration, we often use change of variables to simplify the function we need to integrate.When you replace the original variables - from \((x, y, z)\) to - \((u, v, w)\),it becomes easier to work with, especially when the equation becomes cumbersome. It allows focusing on parameters that may have variables that align with constant borders, making the integration process more tractable.With the provided variables transformation - \(u = x\),- \(v = xy\), and- \(w = 3z\),the original region of integration can be more easily described, thus allowing for a straightforward evaluation of the integral.

The region of integration in multiple integrals is pivotal because it defines where the integration actually takes place in the transformed space. Initially, in this example, the region - \(D\) is described in terms of - inequalities: - \(1 \leq x \leq 2\), - \(0 \leq xy \leq 2\), - and \(0 \leq z \leq 1\).Using the change of variables, we convert these to describe a new region - \(G\) in - \((u, v, w)\)-space.The transformed inequalities are:- \(1 \leq u \leq 2\),- \(0 \leq v \leq 2u\),- \(0 \leq w \leq 3\).This new region clearly outlines the limits for integrating in the new variable space, making the process smoother and avoiding mistakes in limits. These clear bounds for the transformed region are crucial for accurately setting up and evaluating the integral.