1. find the volume of the glass by calculating the integral π(ydy) from 0 to a^2.
2. No it is not correct .The integral f(x) from a to b being < 0 simply means that the total area + and - gives a negative result but the function f(x) might be positive
3. separate the variables and get dy/y =4x/(1+x^2) then integrate....it is easy
4. separate the variables again and find dy/y=x^2dx then integrate and use x=1 ,y=1 to find the constant C of integration.