OCR C4 2007 June — Question 3 6 marks

Exam BoardOCR
ModuleC4 (Core Mathematics 4)
Year2007
SessionJune
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicVolumes of Revolution
TypeVolume with trigonometric functions
DifficultyModerate -0.3 This is a straightforward application of the volumes of revolution formula with a standard trigonometric function. Students need to set up ∫₀^π π(sin x)² dx and use the double angle identity sin²x = (1-cos 2x)/2, which is a routine technique taught explicitly for this type of question. The integration is mechanical once the identity is applied, making this slightly easier than average.
Spec4.08d Volumes of revolution: about x and y axes
3 Find the exact volume generated when the region enclosed between the \(x\)-axis and the portion of the curve \(y = \sin x\) between \(x = 0\) and \(x = \pi\) is rotated completely about the \(x\)-axis.
AnswerMarks Guidance
Volume \(= (k)\int_0^{\pi} \sin^2 x (dx)\)B1 where \(k = \pi, 2\pi\) or \(1\); limits necessary
Suitable method for integrating \(\sin^2 x\)*M1 eg \(\int +/- 1 +/- \cos 2x (dx)\) or single integ by parts & connect to \(\int \sin^2 x (dx)\)
\(\int \sin^2 x (dx) = \frac{1}{2}[1 - \cos 2x (dx)\)A1 or \(- \sin x \cos x + \int \cos^2 x (dx)\)
\(\int \cos 2x (dx) = \frac{1}{2} \sin 2x\)A1 or \(- \sin x \cos x + \int 1 - \sin^2 x (dx)\)
Use limits correctlydep*M1
Volume \(= \frac{1}{2}\pi^2\) WWW Exact answerA1 Beware: wrong working leading to \(\frac{2}{3}\pi^2\) 
Volume $= (k)\int_0^{\pi} \sin^2 x (dx)$ | B1 | where $k = \pi, 2\pi$ or $1$; limits necessary
Suitable method for integrating $\sin^2 x$ | *M1 | eg $\int +/- 1 +/- \cos 2x (dx)$ or single integ by parts & connect to $\int \sin^2 x (dx)$
$\int \sin^2 x (dx) = \frac{1}{2}[1 - \cos 2x (dx)$ | A1 | or $- \sin x \cos x + \int \cos^2 x (dx)$
$\int \cos 2x (dx) = \frac{1}{2} \sin 2x$ | A1 | or $- \sin x \cos x + \int 1 - \sin^2 x (dx)$
Use limits correctly | dep*M1 |
Volume $= \frac{1}{2}\pi^2$ WWW Exact answer | A1 | Beware: wrong working leading to $\frac{2}{3}\pi^2$ | 6

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3 Find the exact volume generated when the region enclosed between the $x$-axis and the portion of the curve $y = \sin x$ between $x = 0$ and $x = \pi$ is rotated completely about the $x$-axis.

\hfill \mbox{\textit{OCR C4 2007 Q3 [6]}}
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