To evaluate the integral using cylindrical coordinates, let's first analyze the problem.

Given:

• The function $f(x, y, z) = z$
• The region of integration is defined by: $x^2 + y^2 \leq 25, \quad z \leq 25$

Step 1: Setting up cylindrical coordinates

In cylindrical coordinates:

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

The region $x^2 + y^2 \leq 25$ describes a circular base of radius 5 in the $xy$-plane. Additionally, $z \leq 25$ defines the height of the region.

Step 2: Setting up the bounds

1. For $r$: $0 \leq r \leq 5$ (radius of the circular base).
2. For $\theta$: $0 \leq \theta \leq 2\pi$ (full rotation around the $z$-axis).
3. For $z$: $0 \leq z \leq 25$ (bounded vertically).

Step 3: Write the integral in cylindrical coordinates

The integral becomes: $\iiint_W f(x, y, z) \, dV = \iiint_W z \, r \, dz \, dr \, d\theta$

Expanding with the limits: $\int_0^{2\pi} \int_0^5 \int_0^{25} z \, r \, dz \, dr \, d\theta$

Step 4: Evaluate the integral

We proceed by integrating step-by-step.

1. Integrate with respect to $z$: $\int_0^{2\pi} \int_0^5 \int_0^{25} z \, r \, dz \, dr \, d\theta = \int_0^{2\pi} \int_0^5 \left[ \frac{z^2}{2} \right]_0^{25} r \, dr \, d\theta$ Evaluating the inner integral: $= \int_0^{2\pi} \int_0^5 \frac{25^2}{2} \, r \, dr \, d\theta$ $= \int_0^{2\pi} \int_0^5 \frac{625}{2} \, r \, dr \, d\theta$

2. Integrate with respect to $r$: $= \int_0^{2\pi} \left[ \frac{625}{2} \frac{r^2}{2} \right]_0^5 \, d\theta$ $= \int_0^{2\pi} \frac{625}{2} \cdot \frac{25}{2} \, d\theta$ $= \int_0^{2\pi} \frac{15625}{4} \, d\theta$

3. Integrate with respect to $\theta$: $= \frac{15625}{4} \cdot 2\pi$ $= \frac{15625 \cdot 2 \cdot \pi}{4}$ $= 7812.5 \pi$

Final Answer:

$\iiint_W f(x, y, z) \, dV = 7812.5 \pi$

Would you like further details or have any questions?

Related Questions:

1. How would the setup change if the region was a cone rather than a cylinder?
2. What is the physical interpretation of this integral if $z$ represents density?
3. How do cylindrical coordinates simplify this problem compared to Cartesian coordinates?
4. How would this integral be evaluated if $f(x, y, z) = x + y + z$?
5. What are some common applications of triple integrals in cylindrical coordinates?

Tip:

When solving problems in cylindrical coordinates, always carefully set up the bounds for $r$, $\theta$, and $z$ based on the geometry of the region.