An aerospace engineer designs a flared component for a high-precision propulsion nozzle. The profile of the component's inner radius rrr (in cm) is modeled by the equation
r=3xx2+5,x≥0 r = \sqrt{\frac{3x}{x^2 + 5}}, \quad x \ge 0 r=x2+53x,x≥0where x x\,x represents the axial distance from the intake in cm.
A section of this component, represented by the finite region RRR, is bounded by the curve, the xxx-axis, and the vertical lines x=5x = \sqrt{5}x=5 and x=35x = 3\sqrt{5}x=35.
The component is formed by rotating the region R R\,R through 2π 2\pi\,2π radians about the xxx-axis to form a solid of revolution.
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Use integration to determine the exact volume of the solid generated. Give your answer in the form alnba \ln balnb, where a a\,a is an irrational number and b b\,b is a prime number.
480 exam-style questions on AQA A Level Maths 1.11 H: Integration, covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.