Prove that $\int_{\delta D}x\ dy$ is area of the $D$ and $\int_{\delta D}y\ dx$ is munis the area of $D.$
Now using Green's Theorem I can prove that $$\int_{\delta D}x\ dy+\int_{\delta D}y\ dx\ =\ 0$$ so the second part is readily done.
But problem is proving that $\int_{\delta D}x\ dy$ is the area. By definition , $$\int_{\delta D}x\ dy=lim_{n\rightarrow \infty}\sum_{i=1}^n (x_{j+1})(y_{j+1}-y_j).$$ Now drawing pictures , and considering $x$ to be a function of $y$, I can see the sum on the right tends to the area under the curve but I cannot write the proof analytically.
Thanks for any help.