13.3#9

This problem on the 13.3 webwork is providing to be a pain. I am unable to figure out how to solve for the answers of b,c and d. I have tried taking a vector between the two points and then tried taking the magnitude of the change in x and y when going from P to Q. Is there something I am missing
when it comes to getting this problem completed and correct

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So, you're missing a basic theoretical fact about the line integrals of gradient vector fields (which is discussed in the textbook and  in the lecture notes.  You are paying a lot of money for these things, when you can't do a problem it is highly recommended to look at them. You would be likely to get an answer quicker than I'll get around to giving it to you, and since time is money you're paying extra for the wait.)  The theoretical fact you need is that the fundamental theorem of calculus applies to the line integrals of gradients: ∫_Γ ∇f∙dr = f(r_2) - f(r_1)     where r_1 and r_2 are the initial and final points of the curve Γ.  This means that your answer is the difference of the values of the function on the initial and starting contour. For example the answer to (b) is 34-38 = -4.