Question

The region bounded by the given curves is rotated about the specified axis. find the volume v of the resulting solid by any method. y = −x2 + 9x − 20, y = 0; about the x-axis

Answer 1

Answer:
`V = (\pi)/(30)`

Step-by-step explanation:

First, we compute the points of intersection of the curves
`y = -x^2 + 9x - 20` and
`y = 0`. Since the equation have y as their left sides,


`-x^2 + 9x - 20 = 0 \\-(x - 5)(x - 4) = 0 \\ x = 5, x = 4`

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` (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` and y =0 intersect at x = 4 and x = 5.

Hence, using equation (1), the volume of the solid is given by:


`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 [ (x^5)/(5) - (9x^4)/(2) + (121x^3)/(3) - 180x^2 + 400x + C \right ]_(4)^(5) \\ \indent \boxed{V = (\pi)/(30)}`