To solve this problem, we need to set up the triple integral in cylindrical coordinates. Here’s the step-by-step solution:

Problem Breakdown

1. Function: The given function $f(x, y, z) = \frac{1}{1156} z \sqrt{x^2 + y^2}$.

   In cylindrical coordinates, $x^2 + y^2 = r^2$, so the function becomes: $f(r, \theta, z) = \frac{1}{1156} z \cdot r$

2. Region: The region is defined by: $x^2 + y^2 \leq z \leq 34 - (x^2 + y^2)$

   In cylindrical coordinates, this translates to: $r^2 \leq z \leq 34 - r^2$

3. Volume Element: In cylindrical coordinates, the volume element $dV$ is $r \, dz \, dr \, d\theta$.

Setting up the Integral

The triple integral $\iiint_W f(x, y, z) \, dV$ becomes: $\iiint_W \frac{1}{1156} z \cdot r \cdot r \, dz \, dr \, d\theta = \frac{1}{1156} \int_0^{2\pi} \int_0^{\sqrt{34}} \int_{r^2}^{34 - r^2} z r^2 \, dz \, dr \, d\theta$

Limits of Integration

1. Theta ($\theta$): From $0$ to $2\pi$ (full rotation around the $z$-axis).
2. Radius ($r$): From $0$ to $\sqrt{34}$, which is the radius that satisfies the boundary $x^2 + y^2 \leq 34$.
3. Z ($z$): For a given $r$, $z$ ranges from $r^2$ to $34 - r^2$.

Integral Setup

The integral is: $\frac{1}{1156} \int_0^{2\pi} \int_0^{\sqrt{34}} \int_{r^2}^{34 - r^2} z r^2 \, dz \, dr \, d\theta$

Solving the Integral

1. Integrate with respect to $z$: $\int_{r^2}^{34 - r^2} z r^2 \, dz = r^2 \left[ \frac{z^2}{2} \right]_{r^2}^{34 - r^2}$ Substituting the limits: $= r^2 \cdot \frac{(34 - r^2)^2 - (r^2)^2}{2}$

2. Simplify and Integrate with respect to $r$ and $\theta$: After evaluating the $z$-integral, you proceed to integrate with respect to $r$ and then $\theta$.

This setup gives you the structure needed to evaluate the integral fully. Would you like me to proceed with the explicit calculations, or do you have any questions about the setup?

Related Questions to Expand Understanding:

1. How do cylindrical coordinates simplify integration in regions with circular symmetry?
2. What would be the setup for this integral if spherical coordinates were used instead?
3. How can one verify the limits of integration in cylindrical coordinates?
4. What are the common applications of cylindrical coordinates in physics or engineering?
5. How does changing to cylindrical coordinates affect the volume element?

Tip

When changing to cylindrical coordinates, always check the bounds carefully to ensure they align with the geometry of the region.