I had this specific task in my math exam and didn't solve it correctly. Also, I, unfortunately, don't have any correct result. So I am asking you, if anyone can solve and explain it to me. I would be super grateful. Sorry for my bad English, it's my second language.
Calculate the line integral where l is upper section of circle $$x^2+y^2=16x$$ form point A(16,0) to point B(0,0)
$$\int_l (e^xsiny-7y) dx + (e^xcosy-7)dy$$
I tried doing this : First, I wrote circle like this: $$ x^2−16x+y^2=0,(x−2\sqrt{2})^2+y^2=2\sqrt{2} $$ Then, I wrote that $$ P=e^xsiny−7y $$ and $$ Q=e^xcosy−7 $$
After that I calculated derivatives $$ \frac{dP}{dy} $$ and $$ \frac{dQ}{dx} $$ , put them back in double integral, using Green's theorem. Meaning I did the following: $\iint_D (\frac{dQ}{dx} - \frac{dP}{dy})\,dx\,dy$
I used the points A and B to define the boundaries, but failed when I got zero as an result.
You have the right idéa! Greens theorem states the if $l$ is a closed curve oriented counter clockwise and $D$ is the area which $l$ encloses then $$\int_{l}P\,dx+Q\,dy=\iint_{D}\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,dx\,dy.$$ In your case the simple closed line $l$ is the $x$-axis and the part of the circle $x^{2}+y^{2}=16x$ which lies above the $x$-axis. We also have $P=e^{x}\sin(y)-7y$ and $Q=e^{x}\cos(y)-7$.
We then calculate the partial derivatives: \begin{align*} \frac{\partial Q}{\partial x} &= e^{x}\cos(y) \\ &\text{and}\\ \frac{\partial P}{\partial y} &= e^{x}\cos(y)-7\end{align*} and therefore $$\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=e^{x}\cos(y)-(e^{x}\cos(y)-7)=7.$$ This is really nice since \begin{align*}\int_{l}(e^{x}\sin(y)-7y)\,dx+(e^{x}\cos(y)-7)\,dy &=\iint_{D}7\,dx\,dy\\ &= 7\times \text{Area}(D).\end{align*}
We can re-write the circles equation as $$(x-8)^{2}+y^{2}=8^{2}$$ which is a circle with middle point $(8,0)$ and radius $8$. This means that $D$ is precisely the half disc that is part of this circle and lies above the $x$-axis. Therefore the area of $D$ is $$\frac{1}{2}(8^{2}\pi)$$ so the answer is $$7\times 32\times\pi= 224\pi$$