The surfaces ρ = 3, ρ = 5, φ = 100°, φ = 130°, z = 3, and z = 4.5 define
a closed surface. Find (a) the enclosed volume;

To find the enclosed volume defined by the surfaces \(\rho = 3\), \(\rho = 5\), \(\phi = 100^\circ\), \(\phi = 130^\circ\), \(z = 3\), and \(z = 4.5\), we will use the cylindrical coordinates system. The volume element in cylindrical coordinates \((\rho, \phi, z)\) is given by:

\[
dV = \rho \, d\rho \, d\phi \, dz
\]

### Step 1: Define the limits of integration

- \(\rho\) varies from 3 to 5.
- \(\phi\) varies from \(100^\circ\) to \(130^\circ\). Convert these angles to radians:

\[
\phi_1 = 100^\circ \times \frac{\pi}{180^\circ} = \frac{5\pi}{9} \, \text{radians}
\]
\[
\phi_2 = 130^\circ \times \frac{\pi}{180^\circ} = \frac{13\pi}{18} \, \text{radians}
\]

- \(z\) varies from 3 to 4.5.

### Step 2: Set up the integral for the enclosed volume

The volume \(V\) is given by the integral:

\[
V = \int_{z=3}^{4.5} \int_{\phi=\frac{5\pi}{9}}^{\frac{13\pi}{18}} \int_{\rho=3}^{5} \rho \, d\rho \, d\phi \, dz
\]

### Step 3: Compute the integral

#### Integrate with respect to \(\rho\):
\[
\int_{\rho=3}^{5} \rho \, d\rho = \left. \frac{\rho^2}{2} \right|_{3}^{5} = \frac{5^2}{2} - \frac{3^2}{2} = \frac{25}{2} - \frac{9}{2} = \frac{16}{2} = 8
\]

#### Integrate with respect to \(\phi\):
\[
\int_{\phi=\frac{5\pi}{9}}^{\frac{13\pi}{18}} d\phi = \left. \phi \right|_{\frac{5\pi}{9}}^{\frac{13\pi}{18}} = \frac{13\pi}{18} - \frac{5\pi}{9} = \frac{13\pi}{18} - \frac{10\pi}{18} = \frac{3\pi}{18} = \frac{\pi}{6}
\]

#### Integrate with respect to \(z\):
\[
\int_{z=3}^{4.5} dz = \left. z \right|_{3}^{4.5} = 4.5 - 3 = 1.5
\]

### Step 4: Multiply the results

Now, multiply the results from each integration step to get the enclosed volume:

\[
V = 8 \times \frac{\pi}{6} \times 1.5 = \frac{12\pi}{6} = 2\pi \, \text{cubic units}
\]

Thus, the enclosed volume is:

\[
V = 2\pi \, \text{cubic units}
\]

Would you like to explore further details or have any questions? Here are 5 related questions to expand your understanding:

1. How does the choice of coordinate system simplify the volume calculation?
2. What is the difference between cylindrical and spherical coordinates?
3. How would the problem change if \(\phi\) varied over a different range?
4. Can you find the surface area of the enclosed volume?
5. How would you set up a similar problem in Cartesian coordinates?

**Tip:** When working with volume integrals in different coordinate systems, always consider the geometry of the problem to choose the most convenient system.

Learn how to calculate the enclosed volume defined by surfaces ρ = 3, ρ = 5, φ = 100°, φ = 130°, z = 3, and z = 4.5 using cylindrical coordinates. Step-by-step solution provided with detailed integration setup and calculations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation