For the region bounded by` y=3-x` , `y=0` , `y=2` , and` x=0` revolved about the line` x=5` , we may apply **Washer method** for the integral application for the volume of a solid.

The** formula for the Washer Method** is:

`V = pi int_a^b [(f(x))^2-(g(x))^2]dx`

or

`V = pi int_a^b [(f(y))^2-(g(y))^2]dy`

where *f as function of the outer radius *

* g as a function of the inner radius*

To determine which form we use, we consider the horizontal rectangular strip representation that is perpendicular to the axis of rotation as shown on the attached image. The given strip has a thickness of "`dy` " which is our clue to use the formula:

`V = pi int_a^b [(f(y))^2-(g(y))^2]dy`

For each radius, we follow the `x_2-x_1` . We have `x_2=5` since it a distance between the axis of rotation and each boundary graph.

For the **inner radius**, we have: `g(y) = 5-(3-y) ` simplified to `g(y)=2+y`

Note: `y_(below)` for the inner radius is based from `y =3-x ` rearrange into `x= 3-y`

For the **outer radius**, we have: `f(y) = 5-0` simplified to `f(y)=5` .

Then the **boundary values of y** is `a=0` and `b =2` .

Then the integral will be:

`V = pi int_0^2 [(5)^2-(2+y)^2]dy`

Expand using the FOIL method on:

`(2+y)^2 = (2+y)(2+y)= 4+4y+y^2 and 5^2=25` .

The integral becomes:

`V = pi int_0^2 [25 -(4+4y+y^2 )]dy`

Simplify:

`V = pi int_0^2 [25 -4-4y-y^2 ]dy`

`V = pi int_0^2 [21-4y-y^2 ]dy`

Apply basic integration property:

`int (u-v-w)dy = int (u)dy-int (v)dy-int (w)dy`

`V = pi [ int_0^2 21dy - int_0^2 4ydy -int_0^2 y^2dy]`

For the integration of `int 21 dy` , we apply basic integration property: `int c dx = cx` .

For the integration of `int_0^2 4ydy` and `int_0^2 y^2dy` , we apply the Power rule for integration: `int y^n dx = y^(n+1)/(n+1)` .

`V = pi [ 21y - 4*y^2/2 -y^3/3]|_0^2`

`V = pi [ 21y - 2y^2 -y^3/3]|_0^2`

Apply the definite integral formula: `int _a^b f(x) dx = F(b) - F(a)` .

`V = pi [ 21(2) - 2(2)^2 -(2)^3/3]-pi [ 21(0) - 2(0)^2 -(0)^3/3]`

`V = pi [ 42 - 8 -8/3]-pi [ 0-0-0]`

`V = pi [ 94/3]-pi[0]`

`V =(94pi)/3` or `98.44` (approximated value)

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