I am looking at the proof of the Green theorem.
$$S_1: y_1=f_1(x), a \leq x \leq b$$ $$S_2: y_2=f_2(x), a \leq x \leq b$$ $$\iint_R \frac{\partial{M}}{\partial{y}}=\int_{f_1(x)}^{f_2(x)} \int_a^b \frac{\partial{M}}{\partial{y}}dxdy=\int_a^b \int_{f_1(x)}^{f_2(x)} \frac{\partial{M}}{\partial{y}} dy dx=\int_a^b [M(x,f_2(x))-M(x,f_1(x))]dx=\int_a^b M(x,f_2(x))-\int_a^b M(x,f_1(x)) dx$$
But...why is $\int_a^b M(x,f_2(x))-\int_a^b M(x,f_2(x)) dx$ equal to $-\int_{S_2} M dx- \int_{S_1} Mdx$ ??
Why is there a minus sign ($-$) , before the integral $\int_{S_2} M dx$??