Question

How do you find the volume of the solid generated by revolving the region bounded by the graphs `xy=6, y=2, y=6, x=6`, about the line x=6?

Answer 1

Please see below.

Explanation:

Find the volume of the solid generated by revolving the region bounded by `xy=6`, `y=2`, `y=6`, and `x=6`, about the line `x=6`.

Here is a graph. The region is in blue. A representative slice (black line segments) has been taken at a `y` value. The thickness is `dy`.

The axis of revolution is the line `x=6` and is indicated with a red circular arrow.

Because the thickness is the differential of `y` (rather than `x`), we'll want the curve in terms of `y`.
`xy=6` if and only if `x=6/y`

When rotated, the radius is the `x` on the right minus the `x` on the left. (greater minus lesser values of `x`)

`r = (6-6/y)`

The volume of the representative disk is

`pi r^2 dy = (6-6/y)^2dy`

`= pi(36-72/y+36/y^2)dy`

`y` varies from `2` to `6`, so the volume of the solid is

`V = pi int_2^6 (36-72/y+36/y^2)dy`

`= pi(156 - 72ln(3))`