I just need someone to confirm that I am right or fix me. Maybe someone will show better ways.
Here's the line integral (second type):
$$\int\limits_C (x+y)\;\mathrm{d}x+(x-y)\;\mathrm{d}y$$
where $C$ is given by ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Theory says that we have to get rid either of all $x$ or all $y$. But it is well known that function that defines ellipse is implicit, so there is no way to express either $x$ or $y$. So I took standard parametrization:
$$\begin{cases}x = a\cos t \\ y = b\sin t \\ 0 \le t \le 2\pi\end{cases}$$
and also the following differentials:
$$\begin{cases} \mathrm{d}x = -a\sin t\,\mathrm{d}t \\ \mathrm{d}y = b\cos t\,\mathrm{d}t \end{cases}$$
After all that it is possible to write down usual definite integral, like:
$$\int_0^{2\pi}(a\cos t+b\sin t)(-a\cos t)+(a\cos t-b\sin t)(b\cos t)\;\mathrm{d}t\ldots$$
The rest should be just easy (get rid of brackets, use linear rule of integration, fracting constants out and so on).
Am I right?
P.S. Also have doubts about $0\le x \le 2\pi$ limits for defenite integral...
the partial of x+y with respect to y =1 = the partial of x-y with respect to x .Therefore the form of the line integral is exact so the integral around the closed curve is 0 .