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.
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.
