Answer
$\frac{94 \pi}{3} $
Work Step by Step
Set-up the integration bounded by the equations $ y=3-x , y=0 , y=2, x=0$
The region is revolved about the line x=5, so the washer method in terms of y will be used.
$ \pi \int^2_0 [(5)^2 - (5-(3-y))^2] dy$ , Setup
$ \pi \int^2_0 [ 25 - (2+y)^2] dy$, Simplify
$ \pi \int^2_0 (-y^2 - 4y +21)dy$
Integrate
$\pi [ -\frac{1}{3} y^3 - 2y^2 +21y]^2_0$
Evaluate the definite integral
$ \pi (-\frac{8}{3} -8 +42) -0$
$ \frac{94\pi}{3} $
