27-28 Use a graph of the vector field F and the curve C to guess whether the line integral of F over C is positive, negative, or zero. Then evaluate the line integral. 27. F(x, y) = (x – y) i + xy j, C is the arc of the circle x? + y² = 4 traversed counter- clockwise from (2, 0) to (0, –2) - -

3. Consider the curve 7(t)=i +(/2 In(t))j +R 1 +-k. a) Find the binormal vector B(1)

6. Consider the vector field F(ar, y) = (8r°y+ e")î+ (Ka + e")3, where K is some constant. (a) (5 points) For which value of K is F the gradient of a function? For this value, find f such that F = Vf. (b) (5 points) For the value of K found in part (a), evaluate fe F dr, where C is the segment of the curve y = a from (0,0) to (1, 1). (c) (5 points) If instead K = 1, use Green's Theorem to evaluate f F dr, where C is the boundary of the triangle with vertices (0,0), (2,0), and (2, 1), traversed counterclockwise.

Evaluate 6x ds where C is the curve y = -7x? from (-1, –7) to (1, –7). Enter the scalar line integral in simplified form and its value, given as an exact answer. Provide your answer below: