From the textbook conclusion formula => reversing the orientation of the curve changed the sign of the line integral.

eg. Evaluate integral sub c of 3xy dx, where c is the line segment joining (0,0) and (1,2) with the given orientation.

a) oriented from (0,0) to (1,2) .

b) oriented from (1,2) to (0,0).

the line integral of a is 4 and the line integral of b is -4.

therefore, reversing the orientation of the curve did change the sign of the line integral in this example.

However, I found another example that is against the above conclusion.

eg) Evaluate the integral sub c of 4x^3 ds where C

a) is the line segment from (-2,-1) to (1,2).

b) is the line segment from (1,2) to (-2,-1).

the answer of a and b are the same ---is -15sqrt(2) .

therefore, this question indicated reversing the orientation of the curve did not change the sign of the line integral.

Why?

Please teach me. Thank you.