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The problem is asking to find the volume of the solid generated by rotating the region bounded by the curves:

\[
y = -x^2 + 7x - 10
\]
and
\[
y = 0
\]

about the x-axis.

### Approach:
This type of problem can be solved using the **disk method**, since we're revolving a region around the x-axis.

The formula for the volume \( V \) of the solid of revolution using the disk method is:

\[
V = \pi \int_a^b [f(x)]^2 \, dx
\]

where:
- \( f(x) \) is the function defining the upper curve (in this case, \( y = -x^2 + 7x - 10 \)).
- \( [a, b] \) are the points where the region is bounded.

#### Step 1: Find the bounds \( a \) and \( b \)
To determine the bounds, set \( y = 0 \) in the equation of the curve \( y = -x^2 + 7x - 10 \) and solve for \( x \):

\[
0 = -x^2 + 7x - 10
\]

Solve this quadratic equation to find the values of \( x \) where the curve intersects the x-axis.

#### Step 2: Set up the integral
Once the bounds are determined, set up the integral as:

\[
V = \pi \int_a^b (-x^2 + 7x - 10)^2 \, dx
\]

I will now solve for the bounds and compute the volume.The volume of the solid generated by rotating the region bounded by the curves \( y = -x^2 + 7x - 10 \) and \( y = 0 \) about the x-axis is:

\[
V = \frac{81\pi}{10}
\]

Would you like further details on the process or have any other questions?

Here are 5 related questions you might find useful:
1. How is the disk method different from the washer method?
2. What are the steps to solve for the volume of a solid rotated about the y-axis?
3. How do you solve a similar problem using cylindrical shells?
4. Can the same method be applied if the region is rotated about a line other than the x-axis?
5. What are the conditions for choosing the appropriate method (disk vs washer vs shell)?

**Tip:** When solving for volumes of revolution, it's crucial to determine the correct bounds from where the curves intersect. Solving the equation for \( y = 0 \) is often the first key step.

Find the volume of the solid generated by rotating the region bounded by the curves y = -x^2 + 7x - 10 and y = 0 about the x-axis.

Find the volume of the solid generated by rotating the region bounded by the curves y = -x^2 + 7x - 10 and y = 0 about the x-axis. Learn how to apply the disk method to solve this problem, suitable for students in Grades 11-12 or college calculus.

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