Ta có đồ thị của hàm số \(y=f\left( x \right)\) và \(y=g\left( x \right)\) cắt nhau tại ba điểm có hoành độ lần lượt là \(-2;\ 1;\ 3\)

\(\begin{array}{l}
\Rightarrow \left\{ \begin{array}{l}
- 8a + 4b - 2c + \frac{3}{4} = 4d - 2e - \frac{3}{4}\\
a + b + c + \frac{3}{4} = d + e - \frac{3}{4}\\
27a + 9b + 3c + \frac{3}{4} = 9d + 3e - \frac{3}{4}
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
4a - 2b + c + 2d - e = \frac{3}{4}\\
a + b + c - d - e = - \frac{3}{2}\\
9a + 3b + c - 3d - e = - \frac{1}{2}
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
4a - 2\left( {b - d} \right) + \left( {c - e} \right) = \frac{3}{4}\\
a + \left( {b - d} \right) + \left( {c - e} \right) = - \frac{3}{2}\\
9a + 3\left( {b - d} \right) + \left( {c - e} \right) = - \frac{1}{2}
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
a = \frac{1}{4}\\
b - d = - \frac{1}{2}\\
c - e = - \frac{5}{4}
\end{array} \right..
\end{array}\)

Dựa vào đồ thị hình vẽ ta có:

\(\begin{align}  & S=\int\limits_{-2}^{1}{\left| f\left( x \right)-g\left( x \right) \right|dx}+\int\limits_{1}^{3}{\left| f\left( x \right)-g\left( x \right) \right|dx} \\ & =\int\limits_{-2}^{1}{\left( a{{x}^{3}}+b{{x}^{2}}+cx+\frac{3}{4}-d{{x}^{2}}-ex+\frac{3}{4} \right)dx}+\int\limits_{1}^{3}{\left( d{{x}^{2}}+ex-\frac{3}{4}-a{{x}^{3}}-b{{x}^{2}}-cx-\frac{3}{4} \right)dx} \\ & =\int\limits_{-2}^{1}{\left( a{{x}^{3}}+\left( b-d \right){{x}^{2}}+\left( c-e \right)x+\frac{3}{2} \right)dx}+\int\limits_{1}^{3}{\left( -a{{x}^{3}}-\left( b-d \right){{x}^{2}}-\left( c-e \right)x-\frac{3}{2} \right)dx} \\ & =\int\limits_{-2}^{1}{\left[ \frac{1}{4}{{x}^{3}}-\frac{1}{2}{{x}^{2}}-\frac{5}{4}x+\frac{3}{2} \right]dx}+\int\limits_{1}^{3}{\left[ -\frac{1}{4}{{x}^{3}}+\frac{1}{2}{{x}^{2}}+\frac{5}{4}x-\frac{3}{2} \right]dx} \\ & =\left. \left[ \frac{{{x}^{4}}}{16}-\frac{{{x}^{3}}}{6}-\frac{5{{x}^{2}}}{8}+\frac{3}{2}x \right] \right|_{-2}^{1}-\left. \left[ \frac{{{x}^{4}}}{16}-\frac{{{x}^{3}}}{6}-\frac{5{{x}^{2}}}{8}+\frac{3}{2}x \right] \right|_{1}^{3} \\ & =-\frac{15}{16}-\frac{3}{2}+\frac{15}{8}+\frac{9}{2}-\left( 5-\frac{13}{3}-5+3 \right)=\frac{253}{48}. \\\end{align}\)

Chọn A.