Calculus ll, I would really appreciate some help checking how I did please

I've been trying to solve this calculus ll exercise and I have been having problems, I wanted to know if you can examine it and tell me what you think, Is it wrong? Is it right?, What am I doing wrong?, How you would do it?, anything helps. Hope you can help and thank you very much in advance

Here is part 1.A of my question:

Here is what I did:

Here is part 1.B of my question:

Here is what I did:

Here is part 1.C of my question:

Here is what I did:

Please tell me how I did, and I just wanted to say thank you for your time and help in advance.

Solution 1:

Part (a) is correct.

Part (b) contains an error; while you correctly added $3$ to the inner radius, you did not do the same to the outer radius. So your volume integral needs to be

$$V = \pi \int_{y=0}^3 (-y^2 + 2y + \color{red}{4})^2 - (-y + 4)^2 \, dy.$$

Part (c) also contains an error. The correct computation for the parabolic arc length is

$$1 + (2-2y)^2 = 1 + 4(y-1)^2 = 1 + 4(y^2 - \color{red}{2}y + 1) = 4y^2 - \color{red}{8}y + 5.$$ Whether the question is asking for $L_1 + L_2$ or just $L_2$ by itself, is a matter of interpretation.