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The problem asks to set up an integral using the method of cylindrical shells for the volume of the solid obtained by rotating the shaded region about the x-axis.

Given:
- The curves are \( y = \sqrt{x} \) and \( y = 20 - x \).
- The region is rotated about the x-axis.

### Cylindrical Shells Method Setup:

The formula for the volume of a solid using cylindrical shells when rotating around the x-axis is:

\[
V = \int_{a}^{b} 2\pi y \cdot \text{radius}(y) \cdot \text{height}(y) \, dy
\]

However, since we are rotating around the x-axis, it is easier to express the integral in terms of \( x \). So, the shell height will be the difference between the two functions, and the radius will be \( y \), because we revolve the region about the x-axis.

The limits of integration are determined by the intersection points of \( y = \sqrt{x} \) and \( y = 20 - x \). 

To find the points of intersection:
\[
\sqrt{x} = 20 - x
\]
Squaring both sides:
\[
x = (20 - x)^2
\]
Expanding:
\[
x = 400 - 40x + x^2
\]
Rearranging:
\[
x^2 - 41x + 400 = 0
\]
Solving this quadratic equation using the quadratic formula:
\[
x = \frac{-(-41) \pm \sqrt{(-41)^2 - 4(1)(400)}}{2(1)}
\]
\[
x = \frac{41 \pm \sqrt{1681 - 1600}}{2}
\]
\[
x = \frac{41 \pm \sqrt{81}}{2}
\]
\[
x = \frac{41 \pm 9}{2}
\]
So,
\[
x = \frac{41 + 9}{2} = 25 \quad \text{or} \quad x = \frac{41 - 9}{2} = 16
\]

Thus, the limits of integration are from \( x = 16 \) to \( x = 25 \).

### Setting up the integral:

- The radius of each shell is \( x \).
- The height of each shell is \( (20 - x) - \sqrt{x} \).

Therefore, the volume integral is:

\[
V = 2\pi \int_{16}^{25} x \left[(20 - x) - \sqrt{x}\right] \, dx
\]

Would you like further details on how to compute this integral?

### Follow-up questions:
1. What would the integral look like if the solid were rotated about the y-axis instead?
2. How would you approach this problem if the region was rotated about a line other than the x- or y-axis?
3. Could we solve the volume using the disk or washer method instead of cylindrical shells? How would the setup differ?
4. What if the functions were different, such as \( y = x^2 \) and \( y = 20 - 2x \)? How would that change the setup?
5. How do we handle situations where the region has more than two curves or boundary conditions?

### Tip:
When using the method of cylindrical shells, remember to carefully consider whether the axis of rotation is parallel to the x- or y-axis, as it changes how you set up the radius and height of the shells.

Set up an integral using the method of cylindrical shells for the volume of the solid obtained by rotating the shaded region about the x-axis.

Learn how to set up an integral to find the volume of a solid by rotating the region between y = sqrt(x) and y = 20 - x about the x-axis using the cylindrical shells method. This example covers integral calculus concepts and finding intersection points for integration limits.

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