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.