DIFFERENTIAL OF A DEFINITE INTEGRAL WITH RESPECT TO A VARIABLE INCLUDED IN THE FUNCTION UNDER THE $\int$ SIGN AND IN THE LIMITS OF INTEGRATION. INTEGRALS OF VARIOUS ORDERS FOR FUNCTIONS OF A SINGLE VARIABLE.

Abstract

Let

$$\begin{aligned} A=\int _{z_0}^Z{f(x, z) dz} \end{aligned}$$

be a definite integral relative to z. If, in this integral, we vary separately and independently, one and the other, the three quantities $Z, z_0, x,$ we will find, by virtue of the formulas in (5) (twenty-sixth lecture) (The first version of the fundamental theorem of Calculus,

$$\begin{aligned} \frac{d}{dx}\int _{x_0}^{x}{f(x) dx}=f(x). \end{aligned}$$

) and of formula (2) (thirty-third lecture), (This second formula is,

$$\begin{aligned} \frac{d}{dy}\int _{x_0}^{x}{ f(x, y) dx } = \int _{x_0}^{x}{ \frac{df(x, y)}{dy} dx.} \end{aligned}$$

Without realizing it, Cauchy is again assuming his function is well behaved. As discussed earlier, the exchange of limits that occurs by reversing the order of integration and differentiation he is taking for granted here is not always allowed.)

$$\begin{aligned} \frac{dA}{dZ}=f(x, Z),&\frac{dA}{dz_0}=-f(x, z_0),&\frac{dA}{dx}=\int _{z_0}^Z{\frac{d f(x, z)}{dx} dz}. \end{aligned}$$