MHB Find the volume by using shell and disk method

[jaychay]

Can you check it for me please that I have done it right or not ?
Thank you in advance.

 

[skeeter]

1st one, your integrand is reversed.

$R= 2-\cos{x}$,
$r = 2-\sqrt{4-x^2}$

2nd ... note the curve equation is incorrect.
should be $x=(y-1)^2+1 \implies y = \sqrt{x-1}+1$ for the upper branch of the parabola

alternatively, you could translate both graphs such that the parabola vertex is at the origin, $x=y^2$, with the line having the equation $y = 2-x$ and rotating the shaded region about $y=-1$