Rotation around y and x axis + shifting the fuction

The first problem states to set up a formula for the volume which u get when the area limited by y=X²+1 and y=5 rotates around y=5.

The second problem states to set up a formula for the volume when the area limited by X=2 and y=(8X)^(1/2 rotates around x=2.

My progress on the first part is F(x)=pi*f(x)^2 integrated from -2 to 2 and I realize that I somehow should change the formula of f(x) so it rotates around y=5. And this is where im stuck. I know the answer but I have problems understanding it entirely. I tought that I would just add + 5 in f(x) which is wrong

On the second part I basically arrive at the same problem. I change it to y variable and get the answer F(y)=pi*f(y)^2 integrated from 0 to 4.

. And then I somehow need to change f(y) so it rotates around 2. And I tought that I just add -2 to the formula in y=(8x)^(1/2) which is incorrect.

I appreciate all theoretical help that I can recive. The reason why I have come to my conclusions is because that if y=X² and I want to shift it two steps to the right then I write y=(X-2)² and if I want to shift it 5 steps up I add +5 so I get y=X²+5. But when I tried to apply this to the volume rotation problems I arrive at the wrong answer.

ps. I was trying to add my integrals from a word pad document but it did not work. How do you guys add youre integrals?