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

 

Explanation:

First, we compute the points of intersection of the curves  and . Since the equation have y as their left sides, 



So, the curves intersect at x = 5 and x = 4. Using the ring method

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

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