Evaluate the triple integral \displaystyle \iiint_{E} (x + 8 y) \, dV where E
is bounded by the parabolic cylinder y=4x^2  and the planes z=6x,y=24x,
and z=0
.

To evaluate the triple integral \(\iiint_{E} (x + 8 y) \, dV\) where \(E\) is the region bounded by the parabolic cylinder \(y = 4x^2\) and the planes \(z = 6x\), \(y = 24x\), and \(z = 0\), we will proceed by setting up the integral in appropriate limits.

### 1. **Determine the bounds for \(x\), \(y\), and \(z\):**

- **Bounds for \(x\):** The region \(E\) is bounded by the parabolic cylinder \(y = 4x^2\) and the plane \(y = 24x\). To find the bounds for \(x\), set \(y = 4x^2\) equal to \(y = 24x\):

  \[
  4x^2 = 24x
  \]

  \[
  4x(x - 6) = 0
  \]

  So, \(x = 0\) or \(x = 6\). Therefore, the bounds for \(x\) are \(0 \leq x \leq 6\).

- **Bounds for \(y\):** For each fixed \(x\) in the interval \([0, 6]\), \(y\) varies between the parabolic cylinder \(y = 4x^2\) and the plane \(y = 24x\). So, the bounds for \(y\) are:

  \[
  4x^2 \leq y \leq 24x
  \]

- **Bounds for \(z\):** The region is bounded by \(z = 0\) (the xy-plane) and \(z = 6x\). So, the bounds for \(z\) are:

  \[
  0 \leq z \leq 6x
  \]

### 2. **Set up the triple integral:**

Given the bounds, the triple integral can be written as:

\[
\iiint_{E} (x + 8y) \, dV = \int_{x=0}^{6} \int_{y=4x^2}^{24x} \int_{z=0}^{6x} (x + 8y) \, dz \, dy \, dx
\]

### 3. **Evaluate the innermost integral (with respect to \(z\)):**

\[
\int_{z=0}^{6x} (x + 8y) \, dz = (x + 8y) \int_{z=0}^{6x} \, dz = (x + 8y) \cdot [z]_{0}^{6x} = (x + 8y) \cdot (6x) = 6x(x + 8y)
\]

### 4. **Evaluate the middle integral (with respect to \(y\)):**

\[
\int_{y=4x^2}^{24x} 6x(x + 8y) \, dy = 6x \int_{y=4x^2}^{24x} (x + 8y) \, dy
\]

Separate the integral:

\[
6x \left[ \int_{y=4x^2}^{24x} x \, dy + \int_{y=4x^2}^{24x} 8y \, dy \right]
\]

Evaluate each integral:

\[
6x \left[ x \cdot [y]_{4x^2}^{24x} + 8 \cdot \frac{y^2}{2} \Big|_{4x^2}^{24x} \right]
\]

Calculate:

\[
6x \left[ x \cdot (24x - 4x^2) + 4 \cdot \left((24x)^2 - (4x^2)^2 \right) \right]
\]

Simplifying:

\[
6x \left[ 20x^2 + 4 \left(576x^2 - 16x^4 \right) \right] = 6x \left( 20x^2 + 2304x^2 - 64x^4 \right)
\]

\[
6x \cdot \left( 2324x^2 - 64x^4 \right) = 13944x^3 - 384x^5
\]

### 5. **Evaluate the outermost integral (with respect to \(x\)):**

\[
\int_{x=0}^{6} \left( 13944x^3 - 384x^5 \right) dx
\]

Evaluate each term:

\[
\frac{13944x^4}{4} - \frac{384x^6}{6} \Big|_0^6
\]

Calculate:

\[
\left(\frac{13944 \cdot 6^4}{4} - \frac{384 \cdot 6^6}{6} \right)
\]

Simplify:

\[
\frac{13944 \cdot 1296}{4} - \frac{384 \cdot 46656}{6} = 349056 - 2985984 = -2636928
\]

### Final Answer:

The value of the triple integral is:

\[
\boxed{-2636928}
\]

Would you like further details on any part of this process?

---

1. What are the steps for setting up triple integrals?
2. How do you determine the bounds for a triple integral?
3. How to evaluate integrals involving parabolic cylinders?
4. Can you explain the limits of integration for a different coordinate system?
5. What is the role of each plane in defining the integration region?
6. How would the process change if the bounds were more complex?
7. Could the order of integration affect the complexity of the problem?
8. How can you verify the final result of a multiple integral?

**Tip:** Always check if converting to cylindrical or spherical coordinates could simplify the bounds and the integrand before proceeding with the integral.

Learn how to evaluate the triple integral \(\iiint_{E} (x + 8 y) \, dV\) where \(E\) is bounded by the parabolic cylinder \(y = 4x^2\) and the planes \(z = 6x, y = 24x\), and \(z = 0\). Detailed step-by-step solution with integral setup, bounds determination, and evaluation. Suitable for college-level mathematics.

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