The traffic flow rate (cars per hour) across an intersection is $r(t)=300+600 t-120 t^{2}$, where $t$ is in hours, and $t=0$ is $6 \mathrm{am}$. How many cars pass through the intersection between $6 \mathrm{am}$ and $9 \mathrm{am}$ ?

Step 1 ：The traffic flow rate (cars per hour) across an intersection is given by the function \(r(t)=300+600 t-120 t^{2}\), where \(t\) is in hours, and \(t=0\) is 6 am.

Step 2 ：We want to find out how many cars pass through the intersection between 6 am and 9 am. This is equivalent to finding the integral of the rate function from 0 to 3 (since 9 am is 3 hours after 6 am).

Step 3 ：We calculate the definite integral of the function \(r(t)=300+600 t-120 t^{2}\) from 0 to 3.

Step 4 ：The result of the integral is 2520.

Step 5 ：Thus, the number of cars that pass through the intersection between 6 am and 9 am is \(\boxed{2520}\).