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

  • Dennis M. CatesEmail author
Chapter

Abstract

Let xy be two independent variables, f(xy) 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 thirteenth 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,
$$\begin{aligned} \frac{d}{dy}\int _{x_0}^{X}{f(x, \, y) \, dx}=\int _{x_0}^{X}{\frac{d \, f(x, \, y)}{dy} \, dx}. \end{aligned}$$

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Sun CityUSA

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