The diagram shows part of the curve $$\(y=\frac{6}{x}\)$$. The points $$\((1,6)\)$$ and $$\((3,2)\)$$ lie on the curve. The shaded region is bounded by the curve and the lines $$\(y=2\)$$ and $$\(x=1\)$$. Find the volume generated when the shaded region is rotated through $$\(360^{\circ}\)$$ about the $$\(\boldsymbol{y}\)$$-axis. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_s20_qp_12 Year：2020 Question No：8(a)
Answer:
Volume \(=\pi \int x^{2} \mathrm{~d} y=\pi \int \frac{36}{y^{2}} \mathrm{~d} y\)
\[
=\pi\left[\frac{-36}{y}\right]
\]
Uses limits 2 to 6 correctly \(\rightarrow(12 \pi)\)
Vol of cylinder \(=\pi \cdot 1^{2} .4\) or \(\int 1^{2} . \mathrm{d} y=[y]\) from 2 to 6
\[
\mathrm{Vol}=12 \pi-4 \pi=8 \pi
\]
\[
=\pi\left[\frac{-36}{y}\right]
\]
Uses limits 2 to 6 correctly \(\rightarrow(12 \pi)\)
Vol of cylinder \(=\pi \cdot 1^{2} .4\) or \(\int 1^{2} . \mathrm{d} y=[y]\) from 2 to 6
\[
\mathrm{Vol}=12 \pi-4 \pi=8 \pi
\]
Knowledge points:
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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