A glass component for a specialized microscope objective is designed as a solid of revolution. The component's profile is created by rotating a specific curve through 360∘ 360^\circ\,360∘ radians about the xxx-axis, where the units are centimetres.
The profile of the glass component is modeled by the equation
h(x)=13(2−x)ex,0≤x≤2 h(x) = \frac{1}{3}(2 - x)e^x, \quad 0 \le x \le 2 h(x)=31(2−x)ex,0≤x≤2Show that the volume, V cm3V \text{ cm}^3V cm3, of the silica glass component is given by
V=K∫02(x2−4x+4)e2x dx V = K \int_{0}^{2} (x^2 - 4x + 4)e^{2x} \, dx V=K∫02(x2−4x+4)e2xdxwhere KKK is a constant to be determined.
Hence, calculate the exact volume of the glass component. Give your answer in the form pπ(eq+r) cm3p\pi(e^q + r) \text{ cm}^3pπ(eq+r) cm3, where p,q,p, q,p,q, and rrr are rational numbers to be found.
