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1.9 Calculus

1.9 Calculus

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Question 44

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​+xx​​arctan(x​)2​

for 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 tan⁡u=x\tan u = \sqrt{x}tanu=x​,

a.

show that the volume VVV is given by

k∫ab1u2 du k \int_{a}^{b} \frac{1}{u^2} \, du k∫ab​u21​du

where k,ak, ak,a and bbb are constants to be found.

[5]
b.

Hence, using algebraic integration, find the exact value of VVV.

[3]
Markscheme

1.9 Calculus Questions

  1. A Level
  2. /Maths
  3. /1.9 Calculus

825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.

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