The rate of airflow R R\,R in litres per second into a pneumatic cylinder is modeled by the function
R(t)=(12−3t)lnt,1≤t≤4 R(t) = (12 - 3t) \ln t, \quad 1 \le t \le 4 R(t)=(12−3t)lnt,1≤t≤4where t t\,t is the time in seconds after the valve opens. The total volume of air VVV (in litres) that enters the cylinder between t=1t = 1t=1 and t=4t = 4t=4 corresponds to the area of the region bounded by the curve R(t)R(t)R(t) and the ttt-axis.
Use the trapezium rule with 5 ordinates to find an estimate for the total volume of air V V\,V that enters the cylinder. Give your answer correct to three significant figures.
Show that the exact volume of air V V\,V is given by
48ln2−24.75 48 \ln 2 - 24.75 48ln2−24.75Fully justify your answer.
