University Calculus- Help

This problem deals with the region in the 1st quadrant bounded by y^2= x, y= x-2, and y= 0. Find the volume of the solid obtained by rotating this region about the line x = 4.

So I know you would have to break it apart and integrate between x = 0 to x = 2 and then from x = 2 to x = 4 and add the two volumes, but I'm having a hard time accounting for the line x = 4 for my radii.

I have a midterm this upcoming Friday and any help on this question would be greatly appreciated! Thank you in advance :)

here you appear to be using " 2π radius height thickness '' ; the radius is 4 - x in both cases

if you use disks then π { large radius² - small radius² } thickness , thickness is dy , y in [0,2]

large is [ 4 - y²] , small is [ 4 - ( y + 2) ]

I think Ted s' approach would make the vol of revolution easier.

You're integrating with respect to y, "thin" washers perpendicular to the vertical axis
x=4, from y=0 to y=2, this integrand: π [ (4 - y²)² - {4 - (y + 2)}² ] dy.

After integrating I was evaluating π [ ((y^5)/5) - 3y³ + 2y² +12y ] and I got 72π/5
[I can't guarantee the answer, but I feel confident it's reasonable.]

I feel more confident abt the answer: I made the volume solid and then cut out the
cone {(1/3)π (2²) 2 } and got the same answer.