问题
如下图所示，橘黄色区域所表示的平面图形是由曲线 $x$ $=$ $x(y)$ 与直线 $y$ $=$ $c$, $y$ $=$ $d$ 以及 $y$ 轴所围成的，那么，该平面图形分别绕 $y$ 轴和 $x$ 轴旋转一周所得的旋转体的体积 $V_{y}$ 与 $V_{x}$ 是多少？
选项
[A]. $\begin{cases} & V_{y} = \pi \int_{c}^{d} x(y) \mathrm{d} y \\ & V_{x} = 2 \pi \int_{c}^{d} y |x(y)| \mathrm{d} y \end{cases}$[B]. $\begin{cases} & V_{y} = \pi \int_{c}^{d} x^{2}(y) \mathrm{d} y \\ & V_{x} = 2 \pi \int_{c}^{d} y \cdot x(y) \mathrm{d} y \end{cases}$
[C]. $\begin{cases} & V_{y} = \pi \int_{c}^{d} x^{2}(y) \mathrm{d} y \\ & V_{x} = 2 \pi \int_{c}^{d} y |x(y)| \mathrm{d} y \end{cases}$
[D]. $\begin{cases} & V_{y} = 2 \pi \int_{c}^{d} x^{2}(y) \mathrm{d} y \\ & V_{x} = \pi \int_{c}^{d} y |x(y)| \mathrm{d} y \end{cases}$
$\begin{cases} & V_{\textcolor{Orange}{y}} = \textcolor{Yellow}{\pi} \textcolor{Green}{\cdot} \int_{\textcolor{cyan}{c}}^{\textcolor{cyan}{d}} \textcolor{Red}{x^{2}(y)} \mathrm{d} y \\ & V_{\textcolor{Orange}{x}} = \textcolor{Yellow}{2} \textcolor{Green}{\cdot} \textcolor{Yellow}{\pi} \int_{\textcolor{cyan}{c}}^{\textcolor{cyan}{d}} \textcolor{Red}{y} \textcolor{green}{\cdot} \textcolor{Red}{|x(y)|} \mathrm{d} y \end{cases}$