Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = ln(x),   y = 0,   x = 2;  
 about the x-axis


To set up the integral for the volume of the solid obtained by rotating the region bounded by the curves \( y = \ln(x) \), \( y = 0 \), and \( x = 2 \) about the x-axis, we will use the **method of disks**.

### Step-by-step setup:
1. **Visualizing the region**:
   - The region is bounded by the curve \( y = \ln(x) \) from \( x = 1 \) to \( x = 2 \), as the natural logarithm function starts at \( x = 1 \) for \( y = 0 \) (since \( \ln(1) = 0 \)).
   - We also have \( y = 0 \), which is the x-axis, and the vertical line \( x = 2 \).

2. **Volume of a disk**:
   When we rotate the region about the x-axis, we use the disk method. The volume of each infinitesimally small disk is given by:
   \[
   dV = \pi \cdot [f(x)]^2 \, dx
   \]
   where \( f(x) \) is the radius of the disk, which is the function \( y = \ln(x) \).

3. **Radius of the disk**:
   The radius of each disk is the distance from the x-axis (line of rotation) to the curve, which is simply the value of \( \ln(x) \).

4. **Setting up the integral**:
   The bounds for \( x \) are from \( x = 1 \) to \( x = 2 \). Thus, the volume \( V \) is given by:
   \[
   V = \pi \int_1^2 [\ln(x)]^2 \, dx
   \]

### Final integral setup:
\[
V = \pi \int_1^2 [\ln(x)]^2 \, dx
\]

This is the integral setup for the volume. You would need to evaluate this integral to find the exact volume, but the problem only asks for the setup.

Would you like further details or explanations?

Here are 5 related questions:
1. How do you apply the method of disks for a different axis of rotation?
2. What if the curve was rotated about the y-axis instead?
3. How does the shell method differ from the disk method in calculating volume?
4. What if the upper bound of \( x \) changed? How would that affect the integral?
5. Can we use numerical methods to evaluate the integral if it's hard to solve analytically?

**Tip:** When setting up an integral for volume, always pay attention to the axis of rotation and choose the method (disks or shells) that makes the setup easiest.

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y = ln(x), y = 0, x = 2; about the x-axis

Set up an integral to find the volume of the solid obtained by rotating the region bounded by y = ln(x), y = 0, and x = 2 about the x-axis using the disk method. Learn step-by-step how to set up the correct integral without evaluation. Suitable for college-level calculus students.

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