Which theorem says that $\displaystyle \int_{f_1(x)}^{f_2(x)}\frac{\partial M}{\partial y}\,dy=M(x,f(x_2))-M(x,f(x_1))$?

Question

I want to explain where this equality comes from. I'm working with a proof of Green's theorem. Thanks very much.

Answer 1

Assuming that $\frac{\partial M}{\partial y}$ is continuous in the second variable (or at least that $M$ is absolutely continuous in the second variable), then this is just the fundamental theorem of calculus applied to the single-variable function $$ g_x(t) = \frac{\partial M}{\partial y}(x,t). $$ Indeed, you can check that if we define $G_x(t) := M(x,t)$, then $G_x'(t) = g_x(t)$, so $$ \int_a^b \frac{\partial M}{\partial y}(x,t)\,dt = \int_a^b g_x(t) \,dt = \int_a^b G_x'(t)\,dt = G_x(b) - G_x(a) = M(x,b) - M(x,a). $$

Answer 2

The fundamental theorem of calculus: $\frac d{\operatorname{dx}}\int_a^x f(t)\operatorname{dt}=f(x)$.