1) I always try to do it both ways, just to emphasize the concepts in my mind.
2) That is not . Please demonstrate your work.
3) You have it set up exactly correctly. Try the integral again.
4) And the other way: . See if you get the same answer.
5) Please learn to observe an exploit symmetries. The limits of your integral could have been [0,2] with the result multiplied by 2.