Example 4 Algebraic Expression
$[\sqrt[3]{1 \times 2 \times 3}]+[\sqrt[3]{2 \times 3 \times 4}]+\cdots+$ $[\sqrt[3]{2000 \times 2001 \times 2002}]=(\quad)$.
(A) 2000000
(B) 2001000
(C) 2002000
(D) 2003001
(2000, "Five Sheep Cup" Mathematics Competition (Grade 9))
[Analysis] Notice that each term in the sum can be expressed as $[\sqrt[3]{k(k+1)(k+2)}]$ (where $k$ is a positive integer). We need to estimate $\sqrt[3]{k(k+1)(k+2)}$.
Since $k^{3}<k(k+1)(k+2)<(k+1)^{3}$,
thus, $k<\sqrt[3]{k(k+1)(k+2)}<k+1$.
Therefore, $[\sqrt[3]{k(k+1)(k+2)}]=k$.
Hence the original expression $=1+2+\cdots+2000=2001000$.