The cross-section of a custom industrial bronze plaque is represented by the region SSS bounded by the lines x=2x = 2x=2, x=12x = 12x=12, the xxx-axis, and the curve
y=8ln(x+5) y = 8 \ln(x + 5) y=8ln(x+5)where all coordinates are measured in centimetres.
Use a single trapezium to calculate an approximation for the area of the cross-section SSS, giving your answer in cm2\text{cm}^2cm2 to two decimal places.
The plaque is manufactured by joining three identical bronze pieces, each with a cross-section equal to region SSS. The bronze sheet has a uniform thickness of 5 mm5 \text{ mm}5 mm and a density of 8.9 g/cm38.9 \text{ g/cm}^38.9 g/cm3.
Use the trapezium rule with six ordinates to find an approximate value for the mass of the complete plaque. Give your answer to the nearest gram.
Without further calculation, explain why: (i) the trapezium-rule mass estimate is an underestimate of the mass based on the exact area under the curve, (ii) rounding the ordinates could make the unrounded trapezium-rule mass estimate an overestimate of the estimate found using exact ordinates.
125 exam-style questions on OCR A Level Maths 1.9 Numerical Methods (A-level only), covering 1.9.1 Locating roots by sign change (A-level only), 1.9.2 Failure of sign change methods (A-level only), 1.9.3 Simple iterative methods (A-level only), 1.9.4 Newton-Raphson method (A-level only), 1.9.5 Failure of iterative methods (A-level only), 1.9.6 Numerical integration (A-level only), and 1.9.7 Numerical methods in context (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.