A precision-engineered acoustic horn is modeled as a solid of revolution. The internal profile of the horn is defined by the curve C C\,C with equation
y=2x+xxarctan(x) y = \frac{2}{\sqrt{\sqrt{x} + x\sqrt{x}} \arctan(\sqrt{x})} y=x+xxarctan(x)2for 13≤x≤1\frac{1}{3} \le x \le 131≤x≤1. The region RRR is bounded by the curve CCC, the xxx-axis, and the vertical lines x=13x = \frac{1}{3}x=31 and x=1x = 1x=1.
The internal volume V V\,V of the horn is formed by rotating the region RRR through 360∘ 360^\circ\,360∘ about the xxx-axis.
Using the substitution tanu=x\tan u = \sqrt{x}tanu=x,
show that the volume VVV is given by
k∫ab1u2 du k \int_{a}^{b} \frac{1}{u^2} \, du k∫abu21duwhere k,ak, ak,a and bbb are constants to be found.
Hence, using algebraic integration, find the exact value of VVV.
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.
