P8.4 Integration as limit of a sum

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

This question was well attempted by the vast majority of learners. In part (a), most responses scored the mark, although it was sometimes unclear if they were using dx or δx but this was condoned for this mark. The most common error was the omission of dx , which resulted in the mark not being awarded. Other errors included omission of the limits, using the incorrect limits (usually 0 as the upper or lower limit), incorrectly retaining the limit notation 2.89 1.44 δ 2 lim d x x x →∞∫ or include both dx and δx to give 2.89 1.44 2 δ d x x x ∫ .

Common mistakes

• limit of a sum is ∫ from 1.44 to 2.89 of 2/√x dx
• must include the dx and the limits (common errors: missing dx, limits 0, keeping lim or δx)

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

This question was well attempted by the vast majority of learners. Part (b) of the question was also usually answered successfully. This question was a show that and included the warning they solutions relying on calculator technology were not acceptable, so it was required that learners showed all steps of their working. Most understood the method and showed appropriate integration followed by correct substitution of limits and correct answer. There were sometimes errors with the coefficient, the most common being 1 2 2 x x rather than 2 1 2 x . Other errors when integrating usually occurred with the power (converting 2 x to 1 2 2x then integrating or differentiating instead of integrating) or the use of ln ax . A small number of responses integrated correctly but then proceeded straight to the final answer without any evidence of substitution of the given limits and so were unable to gain any marks. Those who had the incorrect limits such as using 0 were also unable to score either mark, although transcription errors of 2.89 and1.44 were condoned for the method mark. There were many cases of poor notation such as retaining the integral symbol after integrating and include c + within the calculation but these were condoned on this occasion. 9

Common mistakes

• integrate 2x^(−1/2) to 4√x and show the limits substituted: 4 × 1.7 − 4 × 1.2 = 2
• coefficient errors (x^(1/2) × ½), differentiating instead, ln
• jumping straight to 2 scores 0

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

There were a number of candidates who made no attempt at part (a) despite the guidance to express the given sum as an integral. Where candidates had used the information given in the question the correct answer was relatively easy to deduce. Common errors seen in part (a) were omitting the limits or the dx, and writing dx as dx (although this was condoned). Candidates who had attempted part (a) generally went on to attempt part (b). Most candidates provided a fully correct solution and gained both marks. Where errors were made, this was usually at the last mark. Common errors seen were; • incorrectly integrating the expression to ln x or ln 2x differentiating 2 • x rather than integrating it • leaving the answer as 2ln3 rather than ln9

Common mistakes

• integral of 2/x from 2.1 to 6.3 with limits and dx

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

Q05 proved challenging for many candidates and some candidates were unfamiliar with the notation that was used. Most candidates scored either all three marks or zero marks in this question. There were a substantial number of blank responses. It is noted that section 8.4 (guidance) of the Pure Specification states ‘Recognise 0 f( )d lim f( ) b b x a x a x x x x ’. It is also noted that a similar question had not appeared on the 9MA0 June 2018 papers, the SAMs, the specimen papers or the mock papers. Those candidates who recognised that 9 0 4 lim x x x x was another way of writing 9 4 d x x generally produced a correct solution to score all 3 marks. Only a few made errors in their integration of x or in the application of the limits 4 and 1. Most candidates, however, made no creditable progress. Some applied the trapezium rule with 1, h but did not link their trapezium rule attempt with a stated 9 4 d . x x Other candidates attempted to sum x using integer values from 4 x to 9, x resulting in 4 5 6 ... 9 ( 15.1597...). Some candidates mistakenly interpreted ‘ x ’ as a trigger to differentiate , x with a few of these candidates attempting to differentiate x from first principles. A few candidates applied the incorrect method 9 4 4 5 6 ... 9 d x x to give an answer of 2.49 to 3 significant figures.

Common mistakes

• recognise the limit of the sum as ∫₄⁹ √x dx = 38/3
• unfamiliar notation – many blank or used the trapezium rule or summed √4 + … + √9
• δx is not a cue to differentiate