To evaluate the given integral using a change of variables, we first need to understand the region $D$ and then choose an appropriate transformation that simplifies the integration.

Step 1: Identify the bounds of the region $D$

The region $D$ is bounded by the following planes:

1. $y - x = 0$ (or $y = x$)
2. $y - x = 4$ (or $y = x + 4$)
3. $z - y = 0$ (or $z = y$)
4. $z - y = 1$ (or $z = y + 1$)
5. $z = 0$
6. $z = 7$

Step 2: Choose a suitable change of variables

Given the equations of the planes, we can introduce new variables that align with these planes. Let's define: $u = y - x$ $v = z - y$ $w = z$

This change of variables simplifies the boundaries as follows:

• $u = 0$ (corresponding to $y = x$)
• $u = 4$ (corresponding to $y = x + 4$)
• $v = 0$ (corresponding to $z = y$)
• $v = 1$ (corresponding to $z = y + 1$)
• $w = 0$ to $w = 7$ (corresponding to $z = 0$ to $z = 7$)

Step 3: Express the volume element $dV$ in terms of $u, v, w$

First, let's compute the Jacobian determinant of the transformation:

x &= x(u, v, w), \\ y &= y(u, v, w), \\ z &= z(u, v, w). \end{aligned}$$ From the substitution $$u = y - x$$, $$v = z - y$$, and $$w = z$$, we solve: $$\begin{aligned} y &= u + x, \\ z &= v + y = v + u + x. \end{aligned}$$ This implies: $$\begin{aligned} x &= x, \\ y &= u + x, \\ z &= v + u + x. \end{aligned}$$ Taking partial derivatives to form the Jacobian matrix: $$\frac{\partial(x, y, z)}{\partial(u, v, w)} = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{pmatrix}$$ The determinant of this matrix is $$1$$. Thus, the volume element transforms as: $$dV = |1| \, du \, dv \, dw = du \, dv \, dw$$ ### Step 4: Set up the transformed integral With the new variables, the integral becomes: $$\int_{w=0}^{7} \int_{v=0}^{1} \int_{u=0}^{4} \, du \, dv \, dw$$ ### Step 5: Evaluate the integral Integrating step by step: $$\int_{w=0}^{7} \int_{v=0}^{1} \int_{u=0}^{4} du \, dv \, dw = \int_{w=0}^{7} \int_{v=0}^{1} [u]_{0}^{4} \, dv \, dw = \int_{w=0}^{7} \int_{v=0}^{1} 4 \, dv \, dw$$ $$= \int_{w=0}^{7} [4v]_{0}^{1} \, dw = \int_{w=0}^{7} 4 \, dw = 4w \bigg|_{0}^{7} = 4(7) = 28$$ ### Final Answer: The value of the integral is $$\boxed{28}$$. Would you like further details or have any questions? ### Related Questions: 1. How does the Jacobian determinant affect the integral when using a change of variables? 2. Can the integral's value change if the transformation isn't one-to-one? 3. How do you determine the boundaries after a change of variables? 4. What are the applications of changing variables in double and triple integrals? 5. How would you handle a case where the Jacobian determinant is zero? 6. What are some examples of non-linear change of variables? 7. How do you interpret the geometric meaning of the volume element $$dV$$? 8. What happens if the region $$D$$ was defined by different planes? #### Tip: Always ensure the transformation is one-to-one within the region of integration when applying a change of variables.