Question

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 6. (Round your answer to three decimal places.) xy = 6 y = 2 y = 6 x = 6

Answer 1

Explanation:

Here we have the curve xy =6 bounded by the line y=2, y=6 and x=6.

This region is rotated about x =6

We have to find the volume

Since rotated about vertical line parallel to y axis, shifting y axis to right by 6 units we get

the curve equation as

(6-x)y=6

6y-6xy =6

6xy =6(y-1)

x = (y-1)/y

and limits for y is 2 to y

Volume =
`\pi \int x^2 dy\\= \pi \int( (y-1)/(y))^2 dy\\ = \pi \int 1-2/y +1/y^2 dy\\= \pi (y-2ln y -1/y)`

Substitute limits

Volume =
`\pi ( 4-2 ln 3 - 1/3)\\= \pi (11/3 -2ln 3)`