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