Chapter 4.4, Problem 61P

To determine

The expression $\frac{d}{dt}{\displaystyle {\int }_{x_1\left(t\right)}^{x_2\left(t\right)}f\left(x,t\right)dx}={\displaystyle {\int }_{x_1}^{x_2}dx}+f\left(x_2,t\right)\frac{d{x}_2}{dt}-f\left(x_1,t\right)\frac{d{x}_1}{dt}$.

The relation of the expression $\frac{d}{dt}{\displaystyle {\int }_{x_1\left(t\right)}^{x_2\left(t\right)}f\left(x,t\right)dx}={\displaystyle {\int }_{x_1}^{x_2}dx}+f\left(x_2,t\right)\frac{d{x}_2}{dt}-f\left(x_1,t\right)\frac{d{x}_1}{dt}$ to the time derivative of the total amount of a property in a system.

The relation of the expression $\frac{d}{dt}{\displaystyle {\int }_{x_1\left(t\right)}^{x_2\left(t\right)}f\left(x,t\right)dx}={\displaystyle {\int }_{x_1}^{x_2}dx}+f\left(x_2,t\right)\frac{d{x}_2}{dt}-f\left(x_1,t\right)\frac{d{x}_1}{dt}$ to Reynolds transport theorem.