Using the substitution $u=3x+\frac{3}{2}\sin 2x$, show that

$${\int }_0^{\frac{\pi }{2}}{e}^{3x+\frac{3}{2}\sin 2x}{\cos }^{2}xdx=\frac{1}{6}(e^{\frac{3\pi }{2}}-1)$$

The design of a high-performance aerodynamic component involves a surface generated by rotating a region $R$ through $2\pi$ radians about the $x$-axis. The region $R$ is bounded by the curve with equation

$$y=\sqrt{18}{e}^{\frac{3}{2}x+\frac{3}{4}\sin 2x}\cos x$$

and the coordinate axes in the first quadrant.

Use the result from part (a) to find the volume of the component, giving your answer in simplest form.