find the volume of the solid

Question

Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves y=x, y=0, x=2, and x=4 about the line x=1. 

Answer 1

I think this one is easier using the washer method.

 

The volume element  is   2π (x -1) x   dx  .  The factor (x-1) appears instead of the usual x because the vertical rotation axis is at x =1 not at x =0.     The   integral is from 2 to 4 over x.

 

The result of the integration is   2π [ (4³ - 2³ )/3 - (4² - 2² )/2 ]  = 2π ( 12.6666)