Use shells or split the horizontal washers · source reference

Printed question

Find the volume when the area bounded by $y=(x-1)/(x+1)$, the $x$-axis and $x=2,x=3$ is rotated completely about the $y$-axis. Give your answer to 3 significant figures. [4]

[2011/SAJC/Prelim/II/2(a)]

Printed working

Intersections are $(2,1/3)$ and $(3,1/2)$.

$$y=\frac{x-1}{x+1}\Rightarrow x=\frac{1+y}{1-y}.$$

The source uses a purple cylinder minus blue curved annular piece minus red cylinder:

$$\begin{aligned}V&=\pi(3^2)(1/2)-\pi\int_{1/3}^{1/2}x^2dy-\pi(2^2)(1/3)\\&=\frac{9\pi}2-\pi\int_{1/3}^{1/2}\left(\frac{1+y}{1-y}\right)^2dy-\frac{4\pi}3\\&=\frac{19\pi}6-\pi(1.0159)=6.76\quad\text{units}^3.\end{aligned}$$

Teacher notes

Graph: vertical asymptote $x=-1$, horizontal asymptote $y=1$, bounding lines $x=2,x=3$, levels $1/3,1/2$ and “Rotation about y-axis”. The three coloured solid sketches encode the exact subtraction above, not additional formulas.

Printed answer key

$6.76$ units$^3$.

Source diagrams in native form