问题
若曲线 $L$ 的线密度为 $\rho(x)$, 则曲线 $L$ 的质量 $m$ $=$ $?$选项
[A]. $m$ $=$ $\int_{L}$ $\rho(x)$ $\mathrm{d} y$[B]. $m$ $=$ $\int_{L}$ $\rho(x)$ $\mathrm{d} x$
[C]. $m$ $=$ $\int$ $\rho(x)$ $\mathrm{d} s$
[D]. $m$ $=$ $\int_{L}$ $\rho(x)$ $\mathrm{d} s$
$\begin{cases} & V_{\textcolor{Orange}{y}} = \textcolor{Yellow}{\pi} \textcolor{Green}{\cdot} \int_{\textcolor{cyan}{c}}^{\textcolor{cyan}{d}} \textcolor{Red}{x^{2}(y)} \mathrm{d} y \\ & V_{\textcolor{Orange}{x}} = \textcolor{Yellow}{2} \textcolor{Green}{\cdot} \textcolor{Yellow}{\pi} \int_{\textcolor{cyan}{c}}^{\textcolor{cyan}{d}} \textcolor{Red}{y} \textcolor{green}{\cdot} \textcolor{Red}{|x(y)|} \mathrm{d} y \end{cases}$
$\begin{cases} & V_{\textcolor{Orange}{x}} = \textcolor{Yellow}{\pi} \textcolor{Green}{\cdot} \int_{\textcolor{cyan}{a}}^{\textcolor{cyan}{b}} \textcolor{Red}{y^{2}(x)} \mathrm{d} x \\ & V_{\textcolor{Orange}{y}} = \textcolor{Yellow}{2} \textcolor{Green}{\cdot} \textcolor{Yellow}{\pi} \int_{\textcolor{cyan}{a}}^{\textcolor{cyan}{b}} \textcolor{Red}{x} \textcolor{green}{\cdot} \textcolor{Red}{|y(x)|} \mathrm{d} x \end{cases}$
$S$ $=$ $\int_{\textcolor{Orange}{c}}^{\textcolor{Orange}{d}}$ $\textcolor{Yellow}{|} \textcolor{Red}{\Omega(x)} – \textcolor{cyan}{\Delta(x)} \textcolor{Yellow}{|}$ $\mathrm{d} x$
或者写成:
$S$ $=$ $\int_{\textcolor{Orange}{c}}^{\textcolor{Orange}{d}}$ $\textcolor{Yellow}{|} \textcolor{cyan}{\Delta(x)} – \textcolor{Red}{\Omega(x)} \textcolor{Yellow}{|}$ $\mathrm{d} x$
$S$ $=$ $\int_{\textcolor{Orange}{a}}^{\textcolor{Orange}{b}}$ $\textcolor{Yellow}{|} \textcolor{Red}{f(x)} – \textcolor{cyan}{g(x)} \textcolor{Yellow}{|}$ $\mathrm{d} x$
或者写成:
$S$ $=$ $\int_{\textcolor{Orange}{a}}^{\textcolor{Orange}{b}}$ $\textcolor{Yellow}{|} \textcolor{cyan}{g(x)} – \textcolor{Red}{f(x)} \textcolor{Yellow}{|}$ $\mathrm{d} x$