Find the mass of a thin funnel in the shape of a cone 
z = 

x2 + y2
, 1 ≤ z ≤ 3,
 if its density function is 
𝜌(x, y, z) = 5 − z.
 

To find the mass of a thin funnel in the shape of a cone with the given parameters, we can use the triple integral for mass:

\[
\text{Mass} = \iiint_V \rho(x, y, z) \, dV
\]

### Step 1: Express the equation of the cone
The given equation \( z = x^2 + y^2 \) represents a cone in 3D space. 

### Step 2: Convert to cylindrical coordinates
We can simplify the problem by converting the variables to cylindrical coordinates:

\[
x = r\cos\theta, \quad y = r\sin\theta, \quad z = z
\]

The equation of the cone in cylindrical coordinates is:

\[
z = r^2
\]

### Step 3: Define the density function in cylindrical coordinates
The density function \( \rho(x, y, z) = 5 - z \) becomes \( \rho(r, \theta, z) = 5 - z \).

### Step 4: Set up the limits of integration
The limits for \( z \) are given as \( 1 \leq z \leq 3 \). For each fixed \( z \), the radius \( r \) varies from 0 to \( \sqrt{z} \) (since \( r = \sqrt{z} \) from the cone equation). The angle \( \theta \) varies from 0 to \( 2\pi \).

### Step 5: Express the volume element in cylindrical coordinates
The volume element \( dV \) in cylindrical coordinates is \( r \, dr \, d\theta \, dz \).

### Step 6: Set up the integral for the mass
The mass \( M \) is given by:

\[
M = \int_{\theta=0}^{2\pi} \int_{z=1}^{3} \int_{r=0}^{\sqrt{z}} (5 - z) \, r \, dr \, d\theta \, dz
\]

### Step 7: Evaluate the integral
1. **Integrate with respect to \( r \):**

\[
\int_{r=0}^{\sqrt{z}} (5 - z) \, r \, dr = \left[ \frac{(5 - z)r^2}{2} \right]_{r=0}^{\sqrt{z}} = \frac{(5 - z)z}{2}
\]

2. **Integrate with respect to \( \theta \):**

\[
\int_{\theta=0}^{2\pi} \frac{(5 - z)z}{2} \, d\theta = \frac{(5 - z)z}{2} \cdot 2\pi = \pi z(5 - z)
\]

3. **Integrate with respect to \( z \):**

\[
M = \int_{z=1}^{3} \pi z(5 - z) \, dz
\]

Expand the integrand:

\[
M = \pi \int_{z=1}^{3} (5z - z^2) \, dz
\]

Evaluate the integral:

\[
M = \pi \left[ \frac{5z^2}{2} - \frac{z^3}{3} \right]_{z=1}^{3}
\]

Substitute the limits:

\[
M = \pi \left[\left(\frac{5(3)^2}{2} - \frac{(3)^3}{3}\right) - \left(\frac{5(1)^2}{2} - \frac{(1)^3}{3}\right)\right]
\]

Simplify:

\[
M = \pi \left[\left(\frac{45}{2} - 9\right) - \left(\frac{5}{2} - \frac{1}{3}\right)\right]
\]

\[
M = \pi \left[\frac{27}{2} - \frac{13}{6}\right] = \pi \left[\frac{81}{6} - \frac{13}{6}\right] = \pi \cdot \frac{68}{6} = \frac{34\pi}{3}
\]

Thus, the mass \( M \) of the funnel is:

\[
M = \frac{34\pi}{3}
\]

Would you like more details on any step? Here are some related questions:

1. What are cylindrical coordinates, and how do they simplify solving integrals?
2. How do you convert the equation of a cone into cylindrical coordinates?
3. Why do we use triple integrals to find the mass of a three-dimensional object?
4. Can this approach be used to find the mass of other shapes, like a sphere or cylinder?
5. What is the significance of the density function in calculating mass?

**Tip:** When integrating in cylindrical coordinates, always remember to include the extra factor of \( r \) in the volume element \( dV = r \, dr \, d\theta \, dz \).

Learn how to calculate the mass of a cone-shaped funnel using triple integrals and cylindrical coordinates. Detailed step-by-step solution with integration and density function explanation. Suitable for college-level mathematics and physics students.

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