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.
No big deal here. the difference between the two was that in the first problem, you were integrating with respect to x. for the second problem you were integrating with respect s, the arc length. since arclength only has one sign, regardless of what orientation you are using ( is always positive), you get the same result going either way. but since and can change signs based on orientation, we have that, in general:
and
while