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Date May 2011 Marks available 6 Reference code 11M.1.hl.TZ1.7
Level HL only Paper 1 Time zone TZ1
Command term Find Question number 7 Adapted from N/A

Question

Find the area enclosed by the curve y=arctanx , the x-axis and the line x=3 .

Markscheme

METHOD 1

area=30arctanxdx     A1

attempting to integrate by parts     M1

=[xarctanx]3030x11+x2dx     A1A1

=[xarctanx]30[12ln(1+x2)]30     A1

Note: Award A1 even if limits are absent.

 

=π312ln4     A1

(=π33ln2)

METHOD 2

area=π33π30tanydy     M1A1A1

 =π33+[ln|cosy|]π30     M1A1

=π33+ln12     (=π33ln2)     A1

[6 marks]

Examiners report

Many candidates were able to write down the correct expression for the required area, although in some cases with incorrect integration limits. However, very few managed to achieve any further marks due to a number of misconceptions, in particular arctanx=cotx=cosxsinx. Candidates who realised they should use integration by parts were in general very successful in answering this question. It was pleasing to see a few alternative correct approaches to this question.

Syllabus sections

Topic 6 - Core: Calculus » 6.5 » Area of the region enclosed by a curve and the x-axis or y-axis in a given interval; areas of regions enclosed by curves.
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