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Date	May 2018	Marks available	4	Reference code	18M.1.SL.TZ1.S_5
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 1
Command term	Find	Question number	S_5	Adapted from	N/A

Question

Let $f\left( x \right) = \frac{1}{{\sqrt {2x - 1} }}$ , for $x > \frac{1}{2}$ .

Find $\int {{{\left( {f\left( x \right)} \right)}^2}{\text{d}}x}$ .

[3]

Part of the graph of f is shown in the following diagram.

The shaded region R is enclosed by the graph of f, the x-axis, and the lines x = 1 and x = 9 . Find the volume of the solid formed when R is revolved 360° about the x-axis.

[4]

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

correct working (A1)

eg $\int {\frac{1}{{2x - 1}}{\text{d}}x,\,\,\int {{{\left( {2x - 1} \right)}^{ - 1}},\,\,\frac{1}{{2x - 1}},\,\,\int {{{\left( {\frac{1}{{\sqrt u }}} \right)}^2}\frac{{{\text{d}}u}}{2}} } }$

${\int {\left( {f\left( x \right)} \right)} ^2}{\text{d}}x = \frac{1}{2}{\text{ln}}\left( {2x - 1} \right) + c$ A2 N3

Note: Award A1 for $\frac{1}{2}{\text{ln}}\left( {2x - 1} \right)$ .

[3 marks]

attempt to substitute either limits or the function into formula involving f ² (accept absence of $\pi$ / dx) (M1)

eg $\int_1^9 {{y^2}{\text{d}}x,\,\,} \pi {\int {\left( {\frac{1}{{\sqrt {2x - 1} }}} \right)} ^2}{\text{d}}x,\,\,\left[ {\frac{1}{2}{\text{ln}}\left( {2x - 1} \right)} \right]_1^9$

substituting limits into their integral and subtracting (in any order) (M1)

eg $\frac{\pi }{2}\left( {{\text{ln}}\left( {17} \right) - {\text{ln}}\left( 1 \right)} \right),\,\,\pi \left( {0 - \frac{1}{2}{\text{ln}}\left( {2 \times 9 - 1} \right)} \right)$

correct working involving calculating a log value or using log law (A1)

eg ${\text{ln}}\left( 1 \right) = 0,\,\,{\text{ln}}\left( {\frac{{17}}{1}} \right)$

$\frac{\pi }{2}{\text{ln}}17\,\,\,\,\left( {{\text{accept }}\pi {\text{ln}}\sqrt {17} } \right)$ A1 N3

Note: Full FT may be awarded as normal, from their incorrect answer in part (a), however, do not award the final two A marks unless they involve logarithms.

[4 marks]

Examiners report

[N/A]

Syllabus sections

Topic 5—Calculus » SL 5.5—Integration introduction, areas between curve and x axis

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Topic 5—Calculus

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