A chemical reactor's rate of gas production, R(t)R(t)R(t) in m3/hr\text{m}^3/\text{hr}m3/hr, is monitored over a period of time t t\,t in hours. For the interval 1≤t≤31 \le t \le 31≤t≤3, the production rate is modeled by the function:
R(t)=(6−2t)lnt R(t) = (6 - 2t) \ln t R(t)=(6−2t)lntA graph of this function shows that the production rate starts at zero at t=1t = 1t=1, reaches a maximum, and returns to zero at t=3t = 3t=3.
Use the trapezium rule with 5 ordinates to find an estimate for the total volume of gas produced between t=1t = 1t=1 and t=3t = 3t=3. Give your answer correct to three significant figures.
Use integration to find the exact volume of gas produced and show that it is given by
9ln3−8 9 \ln 3 - 8 9ln3−8Fully justify your answer.
