设函数 $f(x)=\left\{\begin{array}{cc}x^2 & 0 \leq x \leq 1 \\ 2-x & 1 < x \leq 2\end{array}\right.$ ，记$F(x)=\int_0^x f(t) \mathrm{d} t, 0 \leq x \leq 2$, 则
A. $F(x)=\left\{\begin{array}{cl}\frac{x^3}{3}, & 0 \leq x \leq 1 \\ \frac{1}{3}+2 x-\frac{x^2}{2}, & 1 < x \leq 2\end{array}\right.$
B. $F(x)=\left\{\begin{array}{cc}\frac{x^3}{3}, & 0 \leq x \leq 1 \\ -\frac{7}{6}+2 x-\frac{x^2}{2}, 1 < x \leq 2\end{array}\right.$
C. $F(x)=\left\{\begin{array}{cr}\frac{x^3}{3}, & 0 \leq x \leq 1 \\ \frac{x^3}{3}+2 x-\frac{x^2}{2}, 1 < x \leq 2\end{array}\right.$
D. $F(x)=\left\{\begin{array}{c}\frac{x^3}{3}, \quad 0 \leq x \leq 1 \\ 2 x-\frac{x^2}{2}, 1 < x \leq 2\end{array}\right.$