The flow rate of a lubricant, QQQ, in litres per hour, through a high-precision valve is modeled by the function Q(t)=3t3t2+1Q(t) = \frac{3t^3}{t^2+1}Q(t)=t2+13t3, where ttt is the time in hours since the valve was opened for 0≤t≤40 \le t \le 40≤t≤4.
An engineer is attempting to approximate the total volume of lubricant released, V=∫04Q(t) dtV = \int_{0}^{4} Q(t) \, dtV=∫04Q(t)dt, using the trapezium rule by splitting the interval into nnn equal strips.
When n=4n = 4n=4: (i) State the number of ordinates that the engineer uses. (ii) Calculate the approximation for the total volume VVV using this method. Give your answer correct to two decimal places.
Show that the exact volume of lubricant released is 24−32ln1724 - \frac{3}{2}\ln 1724−23ln17 litres. Fully justify your answer.
Explain what would happen to the engineer's approximation in part (a)(ii) as n→∞n \to \inftyn→∞.
Practise AQA A Level Maths 1.12 I: Numerical methods (A-level only) with exam-style questions for A Level Maths. 107 questions covering 1.12.1 Locating roots by change of sign (A-level only), 1.12.2 Iterative methods and Newton-Raphson (A-level only), 1.12.3 Numerical integration (A-level only), and 1.12.4 Numerical methods in context (A-level only), matched to the AQA A Level Maths (7357) 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.
