I am new to calculus and am looking for some feedback regarding the following question. Many thanks in advance!
The following line integral is given: $$\int_C(y + y \cos(xy))dx + (x + x \cos(xy))dy$$ Which of the following statements are correct?
$ 1.~$ If $C=(x-3)^2+(y-4)^2=1$, with clockwise movement, then the integral is $0$.
$ 2.~$ If C is circular arc $x^2+y^2=8$ from point $(2,2)$ to point $(-2,-2)$, then the integral is not $0$.
$3.~$ If $1 \le t \le\frac{π}{2}$ and $C=\mathbf C(t)=t\mathbf i+\frac{π}{t}\mathbf j$, then the integral is $0$.
My interpretation is the following:
I found the potential $φ: yx+\sin(xy)$, which means that the outcome of the integral will be independent of its path, and can be calculated via $φ(x_1,y_1)-φ(x_0,y_0)$.
This yields the following results:
$ 1.~$ always $0$ $ 2.~$ $0$ $3.~$ not $0$, based on $φ(\frac{π}{2},2)-φ(1,π)$.
Thus $1$ is correct and $2$ and $3$ are incorrect.
Any comments will be very much appreciated!