C4 2017 5 Edexcel

The volume of the solid is given by:

\begin{align*}\pi \int_0^{\ln 4} y^2 \mathrm dx & = \pi \int_0^{\ln 4} (e^x + 2e^{-x})^2 \mathrm dx \\ & = \pi \int_0^{\ln 4} (e^{2x} + 4 + 4e^{-2x}) \mathrm dx \\ & = \pi \left[\frac 1 2 e^{2x} + 4x - 2e^{-2x}\right]_0^{\ln 4} \\ & = \pi \left(\frac 1 2 e^{2 \ln 4} + 4 \ln 4 - 2 e^{-2 \ln 4} - \frac 1 2 + 2\right) \\ & = \pi \left(8 + 4\ln 4 - \frac 1 8 - \frac 1 2 + 2\right) \\ & = \pi \left(\frac {75} 8 + 4 \ln 4\right)\end{align*}

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