To evaluate this triple integral over the region $D$ using a change of variables, let's start by understanding the boundaries and transforming them into simpler variables.

Given:

• $D$ is bounded by the planes $y - x = 0$, $y - x = 4$, $z - y = 0$, $z - y = 1$, $z = 0$, and $z = 7$.

Step 1: Define New Variables

To simplify the integration limits, we can introduce new variables:

1. Let $u = y - x$
2. Let $v = z - y$
3. Let $w = z$

In this new coordinate system:

• The boundaries become:
  • $u = 0$ to $u = 4$ (from $y - x = 0$ and $y - x = 4$),
  • $v = 0$ to $v = 1$ (from $z - y = 0$ and $z - y = 1$),
  • $w = 0$ to $w = 7$ (from $z = 0$ and $z = 7$).

Step 2: Find the Jacobian Determinant

To change variables, we need the Jacobian determinant of the transformation from $(x, y, z)$ to $(u, v, w)$. We have: $u = y - x, \quad v = z - y, \quad w = z$

Express $x$, $y$, and $z$ in terms of $u$, $v$, and $w$:

1. From $u = y - x$: $y = x + u$.
2. From $v = z - y$: $z = y + v$.
3. Since $w = z$, we can rewrite $y$ and $x$ in terms of $u$, $v$, and $w$ as follows:
   • $y = w - v$
   • $x = w - v - u$

Now, the Jacobian matrix $J$ of the transformation $(x, y, z) \rightarrow (u, v, w)$ is:

\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{vmatrix} = \begin{vmatrix} -1 & -1 & 1 \\ 1 & -1 & 0 \\ 0 & 1 & 1 \end{vmatrix}$$ Calculating the determinant of $$J$$: $$\det(J) = (-1)(-1)(1) + (-1)(1)(0) + (1)(-1)(1) = 1 + 0 - 1 = 1$$ Thus, the absolute value of the Jacobian determinant is $$|J| = 1$$. ### Step 3: Set Up the Integral in Terms of $$u$$, $$v$$, and $$w$$ Since the Jacobian determinant is 1, the volume element $$dV$$ in terms of the new variables is simply $$du \, dv \, dw$$. The integral becomes: $$\int \int \int_D dV = \int_0^7 \int_0^1 \int_0^4 du \, dv \, dw$$ ### Step 4: Evaluate the Integral Now, we can evaluate the integral: $$\int_0^7 \int_0^1 \int_0^4 du \, dv \, dw = \int_0^7 \int_0^1 (4) \, dv \, dw = \int_0^7 (4 \times 1) \, dw = \int_0^7 4 \, dw = 4w \Big|_0^7 = 4 \times 7 = 28$$ ### Final Answer The value of the integral is: $$\int \int \int_D dV = 28$$ Would you like more details, or do you have any questions? --- ### Related Questions: 1. How would we handle the integral if the Jacobian were not equal to 1? 2. Can we use cylindrical or spherical coordinates for other types of regions? 3. What if one or more boundary planes had nonlinear equations? 4. How would a change in any one of the boundary planes affect the result? 5. How does the Jacobian relate to transformations in higher dimensions? ### Tip: When choosing a change of variables, aim for transformations that simplify the region into rectangular or otherwise straightforward limits.