To solve this integral using cylindrical coordinates, let's go through the conversion and setup of the integral carefully.

Problem Setup

The function given is: $f(x, y, z) = z$ The region is bounded by: $x^2 + y^2 \leq z \leq 4$

Step 1: Converting to Cylindrical Coordinates

In cylindrical coordinates:

• $x = r \cos \theta$
• $y = r \sin \theta$
• $z = z$
• $dV = r \, dr \, d\theta \, dz$

The constraint $x^2 + y^2 \leq z$ becomes: $r^2 \leq z$ or equivalently: $r \leq \sqrt{z}$

Step 2: Setting Up the Limits

From the constraint $r^2 \leq z \leq 4$, we set up the limits for each variable:

• $z$ ranges from $r^2$ to 4 (lower bound determined by $r \leq \sqrt{z}$).
• $r$ ranges from 0 to $2$ (since $r \leq \sqrt{4} = 2$).
• $\theta$ ranges from $0$ to $2\pi$ for full rotation around the z-axis.

Step 3: Setting Up the Integral

In cylindrical coordinates, the integral becomes: $\iiint_W f(x, y, z) \, dV = \int_0^{2\pi} \int_0^2 \int_{r^2}^4 z \cdot r \, dz \, dr \, d\theta$

Step 4: Evaluating the Integral

1. Integrate with respect to $z$: $\int_{r^2}^4 z \cdot r \, dz = r \int_{r^2}^4 z \, dz = r \left[ \frac{z^2}{2} \right]_{r^2}^4 = r \left( \frac{16}{2} - \frac{r^4}{2} \right) = r \left( 8 - \frac{r^4}{2} \right) = 8r - \frac{r^5}{2}$

2. Integrate with respect to $r$: $\int_0^2 \left( 8r - \frac{r^5}{2} \right) \, dr = \left[ 4r^2 - \frac{r^6}{12} \right]_0^2 = \left( 4 \cdot 4 - \frac{64}{12} \right) = 16 - \frac{16}{3} = \frac{48 - 16}{3} = \frac{32}{3}$

3. Integrate with respect to $\theta$: $\int_0^{2\pi} \frac{32}{3} \, d\theta = \frac{32}{3} \cdot 2\pi = \frac{64\pi}{3}$

Final Answer

$\iiint_W f(x, y, z) \, dV = \frac{64\pi}{3}$

Would you like further clarification on any steps or additional details on cylindrical coordinates?

Here are five related questions to deepen understanding:

1. How would the limits change if $x^2 + y^2 \leq z \leq 9$?
2. What if $f(x, y, z) = x + y + z$ instead of $f(x, y, z) = z$?
3. Can you explain why $dV = r \, dr \, d\theta \, dz$ in cylindrical coordinates?
4. How do cylindrical coordinates simplify the integration process in this case?
5. What is the impact of changing the region of integration on the limits for $r$ and $z$?

Tip: When setting up integrals in cylindrical coordinates, always carefully analyze the region to determine the bounds for $r$, $\theta$, and $z$.