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.