To find the volume $V$ of the solid obtained by rotating the region bounded by the curves about the line $x = 1$, we will use the method of cylindrical shells.

Step 1: Set up the problem

The region is bounded by:

• $y = x$
• $y = 0$ (the x-axis)
• $x = 3$
• $x = 6$

We are rotating about the line $x = 1$, so the radius of the cylindrical shells will be the horizontal distance from $x = 1$ to a point on the curve, and the height of the shell will be the value of $y = x$.

Step 2: Formula for cylindrical shells

The volume of the solid obtained by rotating a region about a vertical line (here $x = 1$) is given by the formula:

$V = 2\pi \int_{a}^{b} \text{(radius)} \times \text{(height)} \, dx$

Where:

• The radius of each shell is $|x - 1|$.
• The height of each shell is the function value, $y = x$.
• The limits of integration are from $x = 3$ to $x = 6$.

Step 3: Set up the integral

Here, the radius is $|x - 1| = x - 1$ (since $x$ is always greater than 1 within the given bounds), and the height is $x$.

Thus, the integral becomes:

$V = 2\pi \int_{3}^{6} (x - 1) \cdot x \, dx$

Step 4: Simplify the integral

Simplifying the integrand:

$(x - 1) \cdot x = x^2 - x$

So the integral becomes:

$V = 2\pi \int_{3}^{6} (x^2 - x) \, dx$

Step 5: Compute the integral

Now, let's compute the integral:

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

Evaluating this from $x = 3$ to $x = 6$:

$\left[ \frac{x^3}{3} - \frac{x^2}{2} \right]_{3}^{6}$

At $x = 6$:

$\frac{6^3}{3} - \frac{6^2}{2} = \frac{216}{3} - \frac{36}{2} = 72 - 18 = 54$

At $x = 3$:

$\frac{3^3}{3} - \frac{3^2}{2} = \frac{27}{3} - \frac{9}{2} = 9 - 4.5 = 4.5$

Now subtract:

$54 - 4.5 = 49.5$

Step 6: Final volume

Multiplying by $2\pi$:

$V = 2\pi \times 49.5 = 99\pi$

Thus, the volume of the solid is:

$V = 99\pi \, \text{cubic units}$

Further Details and Questions:

Would you like me to break down the steps further, or is there anything unclear? Here are some related questions to explore:

1. What is the method of disks/washers for solving volume problems, and how does it compare to cylindrical shells?
2. How can we find the volume of a solid when rotating about a horizontal axis, such as $y = 1$?
3. What if the curve was $y = x^2$, how would the process change?
4. How do we handle rotations about slant lines, such as $x = 2x + 1$?
5. How do we use definite integrals in other types of volume problems (e.g., solids of revolution with varying radii)?

Tip: Always carefully choose the correct method of integration (shells vs. washers) depending on the axis of rotation.