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.
Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
