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.
Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
