Water flows into a tank at a rate modelled by
q(t)=te−t/2q(t)=te^{-t/2}q(t)=te−t/2 litres per minute, 0≤t≤40\leq t\leq40≤t≤4,
where t t\,t is measured in minutes. Four rectangles, each of width 1 minute, are used to bound the volume of water entering the tank.
Show that q q\,q is increasing on [0,2][0,2][0,2] and decreasing on [2,4][2,4][2,4].
Write down lower and upper rectangle sums that take account of this change in monotonicity.
Hence calculate lower and upper bounds for the volume, giving each answer to three decimal places.
8 exam-style questions on OCR (MEI) A Level Maths 1.10.7 Upper and lower bounds using rectangles (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.