DIFFERENTIATION AND INTEGRATION UNDER THE $$\int $$ SIGN. INTEGRATION OF DIFFERENTIAL FORMULAS WHICH CONTAIN SEVERAL INDEPENDENT VARIABLES

Let x, y be two independent variables, f(x, y) a function of these two variables, and $$x_0, X$$ two particular values of x. We will find, by setting $$\varDelta y=\alpha \, dy, $$ and employing the notations adopted in the thirteenthIntegration of multiple variable functions lecture, $$\begin{aligned} \varDelta _y\int _{x_0}^{X}{f(x, \, y) \, dx}&=\int _{x_0}^{X}{f(x, \, y+\varDelta y) \, dx}-\int _{x_0}^{X}{f(x, \, y) \, dx} \\&=\int _{x_0}^{X}{\varDelta _y f(x, \, y) \, dx}; \end{aligned}$$ then, in dividing by $$\alpha \, dy, $$ and letting $$\alpha $$ converge toward the limit zero, $$\begin{aligned} \frac{d}{dy}\int _{x_0}^{X}{f(x, \, y) \, dx}=\int _{x_0}^{X}{\frac{d \, f(x, \, y)}{dy} \, dx}. \end{aligned}$$