Water flows into a storage tank at a rate modeled by R(t)=t2e−t/4+4 liters per minute, where t is the time in minutes after the flow begins. The flow is observed for 0≤t≤12.
(a) Provide a definite integral that represents the exact volume of water (in liters) that enters the tank during the 12-minute interval.
(b) Using Simpson’s Rule with n=6 subintervals of equal length, approximate the volume of water that enters the tank. Show the setup of your numerical calculation.
(c) Evaluate the integral from part (a) analytically to obtain the exact volume. Show all steps in your computation.
(d) Compute the absolute and percent error between the approximation from part (b) and the exact value from part (c). Interpret these errors in the context of the tank-filling scenario.