The sides of a bowl are formed by rotating the curve y = 6 ln x, 0 ≤ y ≤ 9, about the y-axis, where x and y are measured in centimetres. The bowl contains water to a height of h cm.

attempt to use $V=\pi \int_{a}^{b} x^{2} \mathrm{d} y$ $x=\mathrm{e}^{\frac{y}{6}}$ or any reasonable attempt to find $x$ in terms of $y$

$V=\pi \int_{0}^{h} \mathrm{e}^{\frac{y}{3}} \mathrm{d} y$

Note: Correct limits must be seen for the $\boldsymbol{A1}$ to be awarded.

$=\pi\left[3 \mathrm{e}^{\frac{y}{3}}\right]_{0}^{h}$

Note: Condone the absence of limits for this A1 mark.

$\begin{aligned} & =3 \pi\left[\mathrm{e}^{\frac{h}{3}}-\mathrm{e}^{0}\right] \\ & =3 \pi\left[\mathrm{e}^{\frac{h}{3}}-1\right] \end{aligned}$

Note: If the variable used in the integral is $x$ instead of $y$ (i.e. $V=\pi \int_{0}^{h} \mathrm{e}^{\frac{x}{3}} \mathrm{d} x$) and the candidate has not stated that they are interchanging $x$ and $y$ then award at most M1M1A0A1A1AG.

maximum volume when $h=9 \mathrm{cm}$ $\max$ volume $=180 \mathrm{cm}^{3}$