Find the volume of the solid inside the surface r² + z² 2 = 4 and outside the surface r = 2 cos θ.

To find the volume of the solid inside the surface \( r^2 + z^2 = 4 \) and outside the surface \( r = 2 \cos \theta \), we'll use cylindrical coordinates \((r, \theta, z)\). 

### Step 1: Understanding the Surfaces

1. **Surface 1: \( r^2 + z^2 = 4 \)**
   - This is a circular cylinder of radius \( 2 \) centered along the \( z \)-axis.

2. **Surface 2: \( r = 2 \cos \theta \)**
   - This is a surface in cylindrical coordinates that forms a cardioid shape in the \( r\theta \)-plane.

### Step 2: Setting up the Boundaries

1. **Limits for \( r \)**:
   - \( r \) ranges from \( 2 \cos \theta \) (outer surface) to \( 2 \) (cylinder).

2. **Limits for \( \theta \)**:
   - \( \theta \) ranges from \( 0 \) to \( 2\pi \).

3. **Limits for \( z \)**:
   - Since \( r^2 + z^2 = 4 \), we have \( z = \pm\sqrt{4 - r^2} \), so \( z \) ranges from \( -\sqrt{4 - r^2} \) to \( \sqrt{4 - r^2} \).

### Step 3: Volume Integral in Cylindrical Coordinates

The volume element in cylindrical coordinates is given by \( dV = r \, dr \, d\theta \, dz \).

The volume \( V \) is:

\[
V = \int_{0}^{2\pi} \int_{2\cos\theta}^{2} \int_{-\sqrt{4 - r^2}}^{\sqrt{4 - r^2}} r \, dz \, dr \, d\theta
\]

### Step 4: Evaluating the Integral

1. **Integral with respect to \( z \)**:
   
   \[
   \int_{-\sqrt{4 - r^2}}^{\sqrt{4 - r^2}} dz = 2\sqrt{4 - r^2}
   \]

2. **Substituting back**:

   \[
   V = \int_{0}^{2\pi} \int_{2\cos\theta}^{2} 2r\sqrt{4 - r^2} \, dr \, d\theta
   \]

Now, let's compute this integral step-by-step.The evaluation of the volume integral took longer than expected due to its complexity. However, we can simplify the process and proceed numerically to get an approximate solution.

Let's break down the problem further or use numerical techniques if needed, ensuring we provide an accurate result. Would you like me to continue with this approach, or would you prefer a different method to find the volume?

Find the volume of the solid inside the surface r² + z² = 4 and outside the surface r = 2 cos θ.

Learn how to calculate the volume of a solid bounded by the cylinder r² + z² = 4 and the cardioid surface r = 2 cos θ. This problem is solved using cylindrical coordinates and involves setting up and evaluating a volume integral. Suitable for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation