设 $f(x)$ 在 $[0,1]$ 连续, 已知
$$
\int_0^1 f(x) \mathrm{d} x=a_0, \int_0^1 x f(x) \mathrm{d} x=a_1, \int_0^1 x^2 f(x) \mathrm{d} x=a_2 .
$$
试计算积分 $\int_0^1\left(\int_0^x\left(\int_0^y f(z) \mathrm{d} z\right) \mathrm{d} y\right) \mathrm{d} x$.
$\text{A.}$
$\text{B.}$
$\text{C.}$
$\text{D.}$