Does the answer for part (c) really have to be 0?

[Differentiate1]

Here's my work typed in Microsoft Word:
http://i.imgur.com/jWPqBDh.png

I have trouble believing the answer is 0 for part c. All I did was use the curl of F from part a and dot it with dr which came out to be 0.

 

[SammyS]

Here's my work typed in Microsoft Word:
http://i.imgur.com/jWPqBDh.png

I have trouble believing the answer is 0 for part c. All I did was use the curl of F from part a and dot it with dr which came out to be 0.

I suspect that in part (c), you are expected to actually carry out the line integration around the closed path C , rather than using the theorem (Storke's Theorem).

(An image of you work follows.)