Q9 Net Change 1 Point Water is put into a storage tank and after 1 minute, the water begins flowing out of a crack.that forms in the bottom of the tank. The water flows out of the tank at a rate of $r(t)$ liters per minute, where \[ r(t)=t^{2} \ln (t), \quad 1 \leq t \leq e \] ( $t$ is measured in minutes). Find the total amount of water (in liters) that flows from the tank between $t=1$ and $t=e$. Give your answer as a decimal number with at least three decimal places. Do not include units. Do not use the letter " $e$ " in your answer.

Step 1 ：The problem is asking for the total amount of water that flows from the tank between \(t=1\) and \(t=e\). This can be found by integrating the rate function \(r(t)\) from \(t=1\) to \(t=e\). The integral of a rate function over an interval gives the total amount of change over that interval. In this case, it will give the total amount of water that flows out of the tank.

Step 2 ：Calculate the integral of \(r(t) = t^{2} \ln (t)\) from \(t=1\) to \(t=e\).

Step 3 ：The total amount of water that flows from the tank between \(t=1\) and \(t=e\) is approximately \(\boxed{4.575}\) liters.