A drinking glass is modelled by rotating the graph of $y = {{\text{e}}^x}$ about the y-axis, for $1 \leqslant y \leqslant 5$ . Find the volume of the glass.

$y = {{\text{e}}^x} \Rightarrow x = \ln y$

volume $= \pi \int_1^5 {{{(\ln y)}^2}{\text{d}}y}$     (M1)A1

using integration by parts     (M1)

$\pi \int_1^5 {{{(\ln y)}^2}{\text{d}}y} = \pi \left[ {y{{(\ln y)}^2}} \right]_1^5 - 2\int_1^5 {\ln y{\text{d}}y}$     A1A1

$= \pi \left[ {y{{(\ln y)}^2} - 2y\ln y + 2y} \right]_1^5$     A1A1

Note: Award A1 marks if $\pi$ is present in at least one of the above lines.

$\Rightarrow \pi \int_1^5 {{{(\ln y)}^2}{\text{d}}y} = \pi {\text{ }}5{(\ln 5)^2} - 10\ln 5 + 8$     A1

[8 marks]

Only the best candidates were able to make significant progress with this question. Quite a few did not consider rotation about the y-axis. Others wrote the correct expression, but seemed daunted by needing to integrate by parts twice.