Subtract the curved core from a cylinder · source reference

Printed question

C: $y=(x^2+5x+3)/(x+1)$. S is bounded by C, the $y$-axis and $y=9/2$. Find the volume when S is rotated completely about the $x$-axis. [3]

[2019/EJC/Prelim/I/6(iii)]

Printed working

The graph labels C, S, $(0,3)$, $x=1$, $y=4.5$, asymptotes $x=-1$ and $y=x+4$.

$$V=\pi(4.5)^2(1)-\pi\int_0^1\left(\frac{x^2+5x+3}{x+1}\right)^2dx=17.516=17.5\quad\text{units}^3\ (3\text{ s.f.})\quad\text{(by GC)}.$$

The calculator screen is represented natively:

Screen item	Printed content
Status row	NORMAL FLOAT AUTO REAL RADIAN MP
Input	$\pi\bigl(4.5^2-\int_0^1((x^2+5x+3)/(x+1))^2dx\bigr)$
Output	17.51610613
Cursor	Block cursor beneath the input/output

Teacher notes

Red region rotated about $x$-axis: cylinder of radius $4.5$ units and height $1$ unit. Green region rotated about $x$-axis: $\pi\int_0^1y^2dx$, where $y=(x^2+5x+3)/(x+1)$. Subtract green from red; the arrows and hatching are retained.

Printed answer key

$17.5$ units$^3$ (3 s.f.).

Source diagrams in native form