The diagram shows part of the curve $$\(y=\frac{8}{x+2}\)$$ and the line $$\(2 y+x=8\)$$, intersecting at points $$\(A\)$$ and $$\(B\)$$. The point $$\(C\)$$ lies on the curve and the tangent to the curve at $$\(C\)$$ is parallel to $$\(A B\)$$. Find the volume generated when the shaded region, bounded by the curve and the line, is rotated through $$\(360^{\circ}\)$$ about the $$\(x\)$$-axis. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_s20_qp_11 Year：2020 Question No：11(b)
Answer:
Volume under line \(=\pi \int\left(-\frac{1}{2} x+4\right)^{2} \mathrm{~d} x=\pi\left[\frac{x^{3}}{12}-2 x^{2}+16 x\right]=(42 \pi)\)
(M1 for volume formula. \(\mathbf{A} \mathbf{2 , 1}\) for integration)
Volume under curve \(=\pi \int\left(\frac{8}{x+2}\right)^{2} \mathrm{~d} x=\pi\left[\frac{-64}{x+2}\right]=(24 \pi)\)
Subtracts and uses 0 to \(6 \rightarrow 18 \pi\)
(M1 for volume formula. \(\mathbf{A} \mathbf{2 , 1}\) for integration)
Volume under curve \(=\pi \int\left(\frac{8}{x+2}\right)^{2} \mathrm{~d} x=\pi\left[\frac{-64}{x+2}\right]=(24 \pi)\)
Subtracts and uses 0 to \(6 \rightarrow 18 \pi\)
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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