I am stuck/unsure about two problems out of my homework (my professor grades homework in my class) and I can't afford to lose points on my final homework assignment. I tried going to the tutoring center at my college but there we no multi-variable calculus tutors available, now I am turning to you guys to ask you to please help me out, pretty please :)!
I have solved the problems and shown my work, but I just need someone to check if they are correct or not because I don't want to lose points on homework. If they are wrong can you please explain to me what I did wrong as well.
Here are the two questions along with my work: http://imgur.com/a/Js8KL
Thank you so much!
Your work looks fine to me.
For the second exercise: I assume they mean the 'open' paraboloid so without the closing cap of $z=1$. I would interpret the question this way and it is also how you solved it.
A small typo: when substituting $z=x^2+y^2$ into $\vec F$, you forgot the square of the $z$-component: $$\vec F = \langle x^2 , y^2 , z^\color{blue}{2}\rangle= \langle x^2 , y^2 , (x^2+y^2)^\color{red}{2}\rangle $$ You also omitted the differential elements before switching to polar coordinates.