Answer:
Explanation:First, we compute the points of intersection of the curves
![y = -x^2 + 9x - 20](https://tex.z-dn.net/?f=y%20%3D%20-x%5E2%20%2B%209x%20-%2020)
and
![y = 0](https://tex.z-dn.net/?f=y%20%3D%200)
. Since the equation have y as their left sides,
![-x^2 + 9x - 20 = 0 \\-(x - 5)(x - 4) = 0 \\ x = 5, x = 4](https://tex.z-dn.net/?f=-x%5E2%20%2B%209x%20-%2020%20%3D%200%0A%5C%5C-%28x%20-%205%29%28x%20-%204%29%20%3D%200%0A%5C%5C%20x%20%3D%205%2C%20x%20%3D%204)
So, the curves intersect at x = 5 and x = 4. Using the ring method
![V = \int_{4}^{5}{\pi(-x^2 + 9x - 20)^2}dx](https://tex.z-dn.net/?f=V%20%3D%20%5Cint_%7B4%7D%5E%7B5%7D%7B%5Cpi%28-x%5E2%20%2B%209x%20-%2020%29%5E2%7Ddx)
(1)
In the ring method, if the region is rotated about x-axis, we use dx in the integration and the independent variable is x. Because the independent variable is x, we use the x-coordinates of points of intersection as the limits of integration.
Note that in equation (1), we use 4 and 5 as limits of integration because the given curves
![y = -x^2 + 9x - 20](https://tex.z-dn.net/?f=y%20%3D%20-x%5E2%20%2B%209x%20-%2020)
and y =0 intersect at x = 4 and x = 5.
Hence, using equation (1), the volume of the solid is given by: