The diagram shows part of the curve with equation $$\(y^{2}=x-2\)$$ and the lines $$\(x=5\)$$ and $$\(y=1\)$$. The shaded region enclosed by the curve and the lines is rotated through $$\(360^{\circ}\)$$ about the $$\(x\)$$-axis. 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Exam No：9709_s21_qp_12 Year：2021 Question No：9
Answer:
Curve intersects \(y=1\) at \((3,1)\)
Volume \(=[\pi] \int(x-2)[\mathrm{d} x]\)
\([\pi]\left[\frac{1}{2} x^{2}-2 x\right]\) or \([\pi]\left[\frac{1}{2}(x-2)^{2}\right]\)
\(=[\pi]\left[\left(\frac{5^{2}}{2}-2 \times 5\right)-\left(\frac{\text { their } 3^{2}}{2}-2 \times\right.\right.\) their 3\(\left.)\right]\)
\(=[\pi]\left[\frac{5}{2}+\frac{3}{2}\right]\) as a minimum requirement for their values
Volume of cylinder \(=\pi \times 1^{2} \times(5-\) their 3\()[=2 \pi]\)
[Volume of solid \(=4 \pi-2 \pi=\rceil 2 \pi\) or \(6.28\)
Alternative method for Question 9
Curve intersects \(y=1\) at \((3,1)\)
Volume of solid \(=\pi \int(x-2)-1[\mathrm{~d} x]\)
\(
\begin{array}{l}
{[\pi]\left[\frac{1}{2} x^{2}-3 x\right] \text { or }[\pi]\left[\frac{1}{2}(x-3)^{2}\right]} \\
=[\pi]\left[\left(\frac{5^{2}}{2}-3 \times 5\right)-\left(\frac{\text { their } 3^{2}}{2}-3 \times \text { their } 3\right)\right]
\end{array}
\)
[Volume of solid \(=4 \pi-2 \pi=] 2 \pi\) or \(6.28\)
Volume \(=[\pi] \int(x-2)[\mathrm{d} x]\)
\([\pi]\left[\frac{1}{2} x^{2}-2 x\right]\) or \([\pi]\left[\frac{1}{2}(x-2)^{2}\right]\)
\(=[\pi]\left[\left(\frac{5^{2}}{2}-2 \times 5\right)-\left(\frac{\text { their } 3^{2}}{2}-2 \times\right.\right.\) their 3\(\left.)\right]\)
\(=[\pi]\left[\frac{5}{2}+\frac{3}{2}\right]\) as a minimum requirement for their values
Volume of cylinder \(=\pi \times 1^{2} \times(5-\) their 3\()[=2 \pi]\)
[Volume of solid \(=4 \pi-2 \pi=\rceil 2 \pi\) or \(6.28\)
Alternative method for Question 9
Curve intersects \(y=1\) at \((3,1)\)
Volume of solid \(=\pi \int(x-2)-1[\mathrm{~d} x]\)
\(
\begin{array}{l}
{[\pi]\left[\frac{1}{2} x^{2}-3 x\right] \text { or }[\pi]\left[\frac{1}{2}(x-3)^{2}\right]} \\
=[\pi]\left[\left(\frac{5^{2}}{2}-3 \times 5\right)-\left(\frac{\text { their } 3^{2}}{2}-3 \times \text { their } 3\right)\right]
\end{array}
\)
[Volume of solid \(=4 \pi-2 \pi=] 2 \pi\) or \(6.28\)
Knowledge points:
1.8.3 evaluate definite integrals (Including simple cases of ‘improper’ integrals, such as)
1.8.4.2 a volume of revolution about one of the axes. (A volume of revolution may involve a region not bounded by the axis of rotation, e.g. the region between and y = 5 rotated about the x-axis.)
Solution:
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