Line integral respect to x , y and length, why the results of -C and C are different beween them?

Question

Integral f(x,y) respect to x along the line C and along the line -C are differnet, but Integral f(x,y) respect to arc length along the lin C and -C are the same. Why does that happen? I dont get it at all

Answer 1

On one hand, if you parametrize your line $C$ by a vectorial function with parameter $t$, you have

$$ \int_C f(x,y) dr = \int_t f(x(t),y(t)) \sqrt{x'(t)^2+y'(t)^2}dt $$

On the other hand, if you are only interested in the integral with respect to $x$, you have

$$ \int_C f(x,y) dx = \int_t f(x(t),y(t)) x'(t)dt $$

Now, lets analyze these integrals. If you integrate the first one along $-C$, the term $\sqrt{x'(t)^2+y'(t)^2}$ will keep the same sign, as $x'(t)$ and $y'(t)$ are squared. On the other hand, for the second one, if you integrate along $-C$, $x'(t)$ may change its sign (depending on the nature of $C$), hence the difference.