Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = x,   y = 0,   x = 3,   x = 7;
  about 
x = 1

To find the volume \( V \) of the solid obtained by rotating the region bounded by the curves \( y = x \), \( y = 0 \), \( x = 3 \), and \( x = 7 \) about the line \( x = 1 \), we'll use the method of cylindrical shells.

### Step 1: Setup for Cylindrical Shells
The method of cylindrical shells is ideal when the region is rotated around a vertical axis. Here, the region is bounded by:

- \( y = x \) (upper curve)
- \( y = 0 \) (lower curve, the x-axis)
- \( x = 3 \) (left boundary)
- \( x = 7 \) (right boundary)

The line of rotation is \( x = 1 \), which is a vertical line.

### Step 2: Formula for Volume Using Shell Method
The volume \( V \) using cylindrical shells is given by the formula:

\[
V = 2\pi \int_{a}^{b} (r(x))(h(x)) \, dx
\]

Where:
- \( r(x) \) is the distance from the axis of rotation (line \( x = 1 \)) to a point on the curve.
- \( h(x) \) is the height of the shell, which is the function \( y = x \).

### Step 3: Define \( r(x) \) and \( h(x) \)
- The height \( h(x) \) is the distance from \( y = 0 \) to \( y = x \), so \( h(x) = x \).
- The radius \( r(x) \) is the horizontal distance from the axis of rotation \( x = 1 \) to a point on the curve. So, \( r(x) = x - 1 \).

### Step 4: Limits of Integration
The region is bounded by \( x = 3 \) and \( x = 7 \), so the limits of integration will be from 3 to 7.

### Step 5: Set Up the Integral
Now, substitute \( r(x) = x - 1 \) and \( h(x) = x \) into the volume formula:

\[
V = 2\pi \int_{3}^{7} (x - 1)(x) \, dx
\]

### Step 6: Solve the Integral
First, expand the integrand:

\[
V = 2\pi \int_{3}^{7} (x^2 - x) \, dx
\]

Now, integrate:

\[
\int (x^2 - x) \, dx = \frac{x^3}{3} - \frac{x^2}{2}
\]

Evaluate from \( x = 3 \) to \( x = 7 \):

\[
\left[ \frac{x^3}{3} - \frac{x^2}{2} \right]_{3}^{7}
\]

Substitute the limits:

For \( x = 7 \):

\[
\frac{7^3}{3} - \frac{7^2}{2} = \frac{343}{3} - \frac{49}{2}
\]

For \( x = 3 \):

\[
\frac{3^3}{3} - \frac{3^2}{2} = \frac{27}{3} - \frac{9}{2}
\]

Now compute each term:

\[
\frac{343}{3} - \frac{49}{2} = \frac{686}{6} - \frac{147}{6} = \frac{539}{6}
\]
\[
\frac{27}{3} - \frac{9}{2} = \frac{54}{6} - \frac{27}{6} = \frac{27}{6}
\]

Finally, subtract the results:

\[
\frac{539}{6} - \frac{27}{6} = \frac{512}{6} = \frac{256}{3}
\]

### Step 7: Multiply by \( 2\pi \)

\[
V = 2\pi \times \frac{256}{3} = \frac{512\pi}{3}
\]

### Final Answer:
The volume \( V \) of the solid is:

\[
V = \frac{512\pi}{3} \, \text{cubic units}.
\]

Would you like further clarification or details on any step?

---

Here are five related questions:
1. How does the shell method differ from the disk/washer method?
2. How would the volume change if the region were rotated around the y-axis instead of \( x = 1 \)?
3. What is the general formula for the volume of solids of revolution?
4. Can you apply the same method for curves that are not functions of \( x \)?
5. How do you determine the axis of rotation when using cylindrical shells?

**Tip:** When rotating around a line other than the coordinate axes, make sure to adjust the radius accordingly to reflect the distance from the curve to the new axis of rotation.

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = x, y = 0, x = 3, x = 7; about x = 1

Learn how to find the volume of a solid of revolution using the method of cylindrical shells. In this example, the region bounded by y = x, y = 0, x = 3, and x = 7 is rotated about the line x = 1. This step-by-step solution demonstrates key concepts in calculus, including setting up and solving the integral for the volume.

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