To evaluate the triple integral $\iiint_E 9z \, dV$, where $E$ is bounded by the cylinder $y^2 + z^2 = 9$ and the planes $x = 0$, $y = 3x$, and $z = 0$ in the first octant, we can follow these steps:

Step 1: Set up the bounds of the integral

1. Cylinder: The equation $y^2 + z^2 = 9$ describes a cylinder with radius 3, extending infinitely along the $x$-axis.
2. Planes: The plane $x = 0$ bounds $E$ on the left, $y = 3x$ bounds $E$ from above, and $z = 0$ bounds it from below.
3. First Octant: We restrict to $x, y, z \geq 0$.

Step 2: Determine the bounds for $x$, $y$, and $z$

1. For $x$: Since $y = 3x$, when $y$ reaches its maximum value of 3 (the radius of the cylinder), we have: $3 = 3x \Rightarrow x = 1.$ Thus, $0 \leq x \leq 1$.

2. For $y$: From the plane $y = 3x$ and the cylinder's projection in the $yz$-plane: $0 \leq y \leq 3.$

3. For $z$: From the cylinder's equation: $0 \leq z \leq \sqrt{9 - y^2}.$

Step 3: Set up the triple integral

The triple integral can be expressed as: $\int_0^1 \int_0^{3x} \int_0^{\sqrt{9 - y^2}} 9z \, dz \, dy \, dx.$

Step 4: Evaluate the integral

1. Integrate with respect to $z$: $\int_0^{\sqrt{9 - y^2}} 9z \, dz = 9 \left[ \frac{z^2}{2} \right]_0^{\sqrt{9 - y^2}} = 9 \cdot \frac{(9 - y^2)}{2} = \frac{9(9 - y^2)}{2}.$

2. Substituting into the next integral: $\int_0^1 \int_0^{3x} \frac{9(9 - y^2)}{2} \, dy \, dx.$

3. Integrate with respect to $y$: [ \int_0^{3x} \frac{9(9 - y^2)}{2} , dy = \frac{9}{2} \left[ 9y - \frac{y^3}{3} \right]_0^{3x} = \frac{9}{2} \left[ 27x - \frac{(3x)^3}{3} \right] = \frac{9}{2} \left[ 27x - 9x^3 \right] = \frac{9}{2} \cdot 9x(3 - x^2) = \frac{81x(3 - x^2)}{2}. ]

4. Now integrate with respect to $x$: [ \int_0^1 \frac{81x(3 - x^2)}{2} , dx = \frac{81}{2} \left[ \frac{3x^2}{2} - \frac{x^4}{4} \right]_0^1 = \frac{81}{2} \left[ \frac{3}{2} - \frac{1}{4} \right] = \frac{81}{2} \left[ \frac{6}{4} - \frac{1}{4} \right] = \frac{81}{2} \cdot \frac{5}{4} = \frac{405}{8}. ]

Final Result

The value of the triple integral is: $\frac{405}{8}.$

If you have any questions or need further details, feel free to ask! Here are some related questions to consider:

1. What other coordinate systems might simplify this integral?
2. How would the bounds change if the cylinder had a different radius?
3. Can you explain how to visualize the region $E$?
4. What happens to the integral if we change the integrand to a different function?
5. How would the result differ if calculated in cylindrical coordinates?