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.
438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.