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Date May 2012 Marks available 3 Reference code 12M.2.sl.TZ1.4
Level SL only Paper 2 Time zone TZ1
Command term Write down Question number 4 Adapted from N/A

Question

The graph of \(y = (x - 1)\sin x\) , for \(0 \le x \le \frac{{5\pi }}{2}\) , is shown below.


The graph has \(x\)-intercepts at \(0\), \(1\), \( \pi\) and \(k\) .

Find k .

[2]
a.

The shaded region is rotated \(360^\circ \) about the x-axis. Let V be the volume of the solid formed.

Write down an expression for V .

[3]
b.

The shaded region is rotated \(360^\circ \) about the x-axis. Let V be the volume of the solid formed.

Find V .

[2]
c.

Markscheme

evidence of valid approach     (M1)

e.g. \(y = 0\) , \(\sin x = 0\)

\(2\pi  = 6.283185 \ldots \)

\(k = 6.28\)     A1     N2

[2 marks]

a.

attempt to substitute either limits or the function into formula     (M1)

(accept absence of \({\rm{d}}x\) )

e.g. \(V = \pi \int_\pi ^k {{{(f(x))}^2}{\rm{d}}x} \) , \(\pi \int {{{((x - 1)\sin x)}^2}} \) , \(\pi \int_\pi ^{6.28 \ldots } {{y^2}{\rm{d}}x} \)

correct expression     A2     N3

e.g. \(\pi \int_\pi ^{6.28} {{{(x - 1)}^2}{{\sin }^2}x{\rm{d}}x} \) , \(\pi \int_\pi ^{2\pi } {{{((x - 1)\sin x)}^2}{\rm{d}}x} \) 

[3 marks]

b.

\(V = 69.60192562 \ldots \)

\(V = 69.6\)     A2     N2

[2 marks]

c.

Examiners report

Candidates showed marked improvement in writing fully correct expressions for a volume of revolution. Common errors of course included the omission of dx , using the given domain as the upper and lower bounds of integration, forgetting to square their function and/or the omission of \(\pi \) . There were still many who were unable to use their calculator successfully to find the required volume.

a.

Candidates showed marked improvement in writing fully correct expressions for a volume of revolution. Common errors of course included the omission of dx , using the given domain as the upper and lower bounds of integration, forgetting to square their function and/or the omission of \(\pi \) . There were still many who were unable to use their calculator successfully to find the required volume.

b.

Candidates showed marked improvement in writing fully correct expressions for a volume of revolution. Common errors of course included the omission of  dx, using the given domain as the upper and lower bounds of integration, forgetting to square their function and/or the omission of \(\pi \) . There were still many who were unable to use their calculator successfully to find the required volume.

c.

Syllabus sections

Topic 2 - Functions and equations » 2.2 » The graph of a function; its equation \(y = f\left( x \right)\) .
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