We are asked to use the shell method to find the volume of the solid formed by revolving the region enclosed by the curves:

$y = 6 - x, \quad y = 0, \quad x = 7$ around the x-axis.

Shell Method Overview:

The shell method revolves vertical slices around the axis of rotation. For rotation around the x-axis, the volume is given by:

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

• Radius of the shell: The distance from the x-axis, which is $y$.
• Height of the shell: The horizontal distance, which is along the x-axis, is represented by the function $x$ as the height depends on the curve $y = 6 - x$.
• Limits of integration: From $y = 0$ to $y = 6$ since the curve starts at $y = 0$ and ends at $y = 6$.

We will express $x$ in terms of $y$, from the equation $y = 6 - x$, we have $x = 6 - y$.

Defining the Integral:

Now applying the shell method, we write the integral for the volume:

$V = 2\pi \int_{0}^{6} y \cdot (7 - (6 - y)) \, dy$ Simplifying the height expression:

$V = 2\pi \int_{0}^{6} y \cdot (1 + y) \, dy$

Solving the Integral:

We can now expand the integrand:

$V = 2\pi \int_{0}^{6} (y + y^2) \, dy$

Now, solve the integral:

$V = 2\pi \left[ \frac{y^2}{2} + \frac{y^3}{3} \right]_{0}^{6}$

Substitute the limits:

$V = 2\pi \left[ \frac{6^2}{2} + \frac{6^3}{3} \right]$

$V = 2\pi \left[ \frac{36}{2} + \frac{216}{3} \right]$

$V = 2\pi \left[ 18 + 72 \right]$

$V = 2\pi \times 90 = 180\pi$

Final Answer:

The volume of the solid generated by revolving the region around the x-axis is $\boxed{180\pi}$.

Would you like more details on the shell method or this solution?

Here are some related questions to deepen your understanding:

1. How does the shell method differ from the disk/washer method?
2. What if the region were rotated around the y-axis instead of the x-axis?
3. How does the choice of axis of rotation affect the setup of the integral?
4. Can we use the disk method to solve this problem? How would it be set up?
5. What are the advantages of using the shell method in certain problems?

Tip: Always verify whether you need to solve for $x$ in terms of $y$ (or vice versa) when setting up shell method integrals, as this depends on the axis of revolution.