An aerospace engineer is designing a prototype for a high-pressure propulsion nozzle. The internal profile of the nozzle's expansion chamber can be modelled by the curve with equation
g(x)=13(2−x)e2x,0≤x≤2 g(x) = \frac{1}{3}(2 - x)e^{2x}, \quad 0 \le x \le 2 g(x)=31(2−x)e2x,0≤x≤2where xxx is the distance in centimetres from the inlet. The 3D shape of the chamber is formed by rotating this curve through 360∘360^\circ360∘ about the xxx-axis.
Show that the internal volume, V cm3V\text{ cm}^3V cm3, of the chamber is given by
V=K∫02(x2−4x+4)e4x dx V = K \int_{0}^{2} (x^2 - 4x + 4)e^{4x} \, dx V=K∫02(x2−4x+4)e4xdxwhere KKK is a constant to be found.
Hence, find the exact value of the volume of the nozzle chamber. Give your answer in the form pπ(eq+r) cm3p\pi(e^q + r)\text{ cm}^3pπ(eq+r) cm3 where p,qp, qp,q and rrr are rational numbers to be found.
864 exam-style questions on Edexcel A Level Maths Integration, 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. Each one has a worked solution and a mark scheme showing where the marks go.