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What is the difference between ∫ F∙dr and ∫ ∇f.dr, line integrals problem?

I'm trying to solve this problem

Attachment 21652

and I got the length by substituting dx by dy or the opposite, or I could use the parametric equation.

But the 2nd part is where I stop, please help me I have an exam in few hours :/ can someone explain in details how I would solve part B? Sorry for my english, not my native.

I have an exam tomorrow, please it is urgent.

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