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To express the integral \(\iiint_E f(x, y, z) \, dV\) as an iterated integral in six different ways, we need to analyze the boundaries of the region \( E \), which is bounded by the following surfaces:

1. \( z = 0 \): The \( z \)-axis plane.
2. \( x = 0 \): The \( x \)-axis plane.
3. \( z = y - 7x \): A plane in the \( x\)-\( y\)-\( z\) space.
4. \( y = 21 \): A plane parallel to the \( z \)-axis.

### Step 1: Analyzing Bounds
The inequalities derived from the given boundaries are:
- \( z \) ranges from \( 0 \) to \( y - 7x \).
- \( x \) ranges from \( 0 \) to some maximum \( x \)-value determined by \( y \).
- \( y \) ranges from \( 0 \) to \( 21 \).

From the condition \( z = y - 7x \), we find that \( y \geq 7x \), which leads to \( x \leq \frac{y}{7} \).

### Step 2: Setting Up the Iterated Integrals
Now we can write the six different iterated forms by choosing different orders for \( x \), \( y \), and \( z \) as the integration variables. Here are the possible setups:

#### 1. \( dz \, dx \, dy \)
\[
\iiint_E f(x, y, z) \, dV = \int_{0}^{21} \int_{0}^{\frac{y}{7}} \int_{0}^{y - 7x} f(x, y, z) \, dz \, dx \, dy
\]

#### 2. \( dz \, dy \, dx \)
\[
\iiint_E f(x, y, z) \, dV = \int_{0}^{3} \int_{0}^{7x} \int_{0}^{y - 7x} f(x, y, z) \, dz \, dy \, dx
\]

#### 3. \( dx \, dz \, dy \)
\[
\iiint_E f(x, y, z) \, dV = \int_{0}^{21} \int_{0}^{y - z} \int_{0}^{\frac{y}{7}} f(x, y, z) \, dx \, dz \, dy
\]

#### 4. \( dx \, dy \, dz \)
\[
\iiint_E f(x, y, z) \, dV = \int_{0}^{21} \int_{0}^{\frac{y}{7}} \int_{0}^{y - 7x} f(x, y, z) \, dy \, dz \, dx
\]

#### 5. \( dy \, dz \, dx \)
\[
\iiint_E f(x, y, z) \, dV = \int_{0}^{3} \int_{0}^{21} \int_{0}^{y - 7x} f(x, y, z) \, dy \, dz \, dx
\]

#### 6. \( dy \, dx \, dz \)
\[
\iiint_E f(x, y, z) \, dV = \int_{0}^{3} \int_{0}^{\frac{21}{7}} \int_{0}^{y - 7x} f(x, y, z) \, dy \, dx \, dz
\]

These six representations cover all possible orders of integration for \(x\), \(y\), and \(z\) over the region defined by \( E \).

Would you like further clarification on any of these setups or a breakdown of any specific iterated integral?

### Related Questions

1. How do you determine the bounds for a triple integral given different constraints?
2. What are some strategies for evaluating complex triple integrals?
3. How does changing the order of integration affect the difficulty of evaluating the integral?
4. Can this method be extended to regions defined by more complex surfaces?
5. What are some practical applications of triple integrals in physics and engineering?

**Tip:** When setting up iterated integrals, always start by analyzing each boundary condition to clearly understand the limits of integration for each variable.

Express the integral \(\iiint_E f(x, y, z) \, dV\) as an iterated integral in six different ways, where \( E \) is the solid bounded by \( z = 0 \), \( x = 0 \), \( z = y - 7x \), and \( y = 21 \).

Learn how to express the triple integral \(\iiint_E f(x, y, z) \, dV\) as an iterated integral in six different ways. The region \( E \) is bounded by planes \( z = 0 \), \( x = 0 \), \( z = y - 7x \), and \( y = 21 \). This problem involves concepts in multivariable calculus, such as triple integrals and bounded regions.

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