Hydroelectric Reservoir Net Change (Civil & Environmental Engineering)

Tracking net water volume changes in hydroelectric reservoirs requires integrating the net flow rate (inflow minus spillway discharge) over an operational time period.

Water discharges through a spillway of a reservoir at a time-varying rate of $r(t) = 18 - \frac{t^2}{2}$ cubic meters per minute for $t \ge 0$, where $t$ is measured in minutes. Initially, the reservoir has $65$ cubic meters of active storage.

Determine the following:

1. Write the definite integral representing the total volume of water discharged (lost) from the reservoir between $t = 0$ and $t = 3$ minutes.
2. Evaluate this integral and determine how much water is remaining in the reservoir after $3$ minutes.