Answer: [tex]V = \frac{\pi}{30}[/tex]
Explanation:
First, we compute the points of intersection of the curves [tex]y = -x^2 + 9x - 20[/tex] and [tex]y = 0[/tex]. Since the equation have y as their left sides,
[tex]-x^2 + 9x - 20 = 0
\\-(x - 5)(x - 4) = 0
\\ x = 5, x = 4[/tex]
So, the curves intersect at x = 5 and x = 4. Using the ring method
[tex]V = \int_{4}^{5}{\pi(-x^2 + 9x - 20)^2}dx[/tex] (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 [tex]y = -x^2 + 9x - 20[/tex] and y =0 intersect at x = 4 and x = 5.
Hence, using equation (1), the volume of the solid is given by:
[tex]V = \int_{4}^{5}{\pi(-x^2 + 9x - 20)^2}dx
\\ \indent = \pi\int_{4}^{5}{(x^4 -18x^3 + 121x^2 − 360x + 400)}dx
\\ \indent = \pi\left [ \frac{x^5}{5} - \frac{9x^4}{2} + \frac{121x^3}{3} - 180x^2 + 400x + C \right ]_{4}^{5}
\\ \indent \boxed{V = \frac{\pi}{30}}[/tex]
