Consider the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis.
y	 = 	25 − x2
y	 = 	0
x	 = 	2
x	 = 	5

To find the volume of the solid generated by revolving the region bounded by the given equations about the y-axis, we'll use the method of cylindrical shells.

### Step 1: Set up the problem
The region is bounded by the following equations:
- \( y = 25 - x^2 \) (a downward-facing parabola)
- \( y = 0 \) (the x-axis)
- \( x = 2 \) and \( x = 5 \) (vertical lines)

### Step 2: Visualize the solid
We are revolving the region around the y-axis. Since we're using the method of cylindrical shells, the formula for the volume \( V \) is:

\[
V = 2\pi \int_{a}^{b} (radius)(height) \, dx
\]

Here:
- The **radius** of a shell is the distance from the y-axis, which is simply \( x \).
- The **height** of a shell is the function \( y = 25 - x^2 \).

### Step 3: Apply the formula
The bounds for \( x \) are from \( x = 2 \) to \( x = 5 \), so:

\[
V = 2\pi \int_{2}^{5} x(25 - x^2) \, dx
\]

### Step 4: Simplify the integrand
Expand the integrand:

\[
V = 2\pi \int_{2}^{5} (25x - x^3) \, dx
\]

### Step 5: Integrate
Now, compute the integral:

\[
\int (25x - x^3) \, dx = \frac{25x^2}{2} - \frac{x^4}{4}
\]

Evaluate this from \( x = 2 \) to \( x = 5 \):

\[
V = 2\pi \left[ \left(\frac{25(5)^2}{2} - \frac{(5)^4}{4}\right) - \left(\frac{25(2)^2}{2} - \frac{(2)^4}{4}\right) \right]
\]

### Step 6: Calculate the definite integral
Substitute the values:

For \( x = 5 \):
\[
\frac{25(5)^2}{2} - \frac{(5)^4}{4} = \frac{25(25)}{2} - \frac{625}{4} = \frac{625}{2} - \frac{625}{4} = \frac{1250}{4} - \frac{625}{4} = \frac{625}{4} = 156.25
\]

For \( x = 2 \):
\[
\frac{25(2)^2}{2} - \frac{(2)^4}{4} = \frac{25(4)}{2} - \frac{16}{4} = \frac{100}{2} - 4 = 50 - 4 = 46
\]

Thus:

\[
V = 2\pi \left(156.25 - 46\right) = 2\pi(110.25) = 220.5\pi
\]

### Final Answer:
The volume of the solid generated by revolving the region about the y-axis is \( \boxed{220.5\pi} \) cubic units.

Would you like more details on any step, or do you have any questions?

Here are 5 related questions to explore further:
1. How would the volume change if the region were revolved around the x-axis instead of the y-axis?
2. What is the volume if the parabola was \( y = 36 - x^2 \) instead?
3. How does the method of disks differ from the method of cylindrical shells for this problem?
4. How would the volume change if the region was revolved around the line \( x = 1 \) instead of the y-axis?
5. Can you calculate the surface area of the solid formed by this revolution?

**Tip:** Always sketch the region and the axis of rotation to better visualize the solid and choose the most suitable method for finding the volume.

Learn how to calculate the volume of a solid formed by revolving the region bounded by the graphs y = 25 - x^2, y = 0, x = 2, and x = 5 about the y-axis using the cylindrical shells method. Detailed step-by-step solution provided.

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