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1.11 H: Integration

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Question 223

The work done WWW by a magnetic force on a micro-particle is determined by its displacement sss (in mm). For 0≤s≤20 \le s \le 20≤s≤2, the work required is given by the integral:

W=∫023s+4(16−s2)32 ds W = \int_{0}^{2} \frac{3s+4}{(16-s^2)^{\frac{3}{2}}} \, ds W=∫02​(16−s2)23​3s+4​ds
a.

Use the substitution s=4sin⁡θs = 4 \sin \thetas=4sinθ to show that

∫023s+4(16−s2)32 ds=∫0p(34sec⁡θtan⁡θ+14sec⁡2θ) dθ \int_{0}^{2} \frac{3s+4}{(16-s^2)^{\frac{3}{2}}} \, ds = \int_{0}^{p} \left( \frac{3}{4} \sec \theta \tan \theta + \frac{1}{4} \sec^2 \theta \right) \, d\theta ∫02​(16−s2)23​3s+4​ds=∫0p​(43​secθtanθ+41​sec2θ)dθ

where ppp is a constant to be found.

[5]
b.

Hence find the exact value of

∫023s+4(16−s2)32 ds \int_{0}^{2} \frac{3s+4}{(16-s^2)^{\frac{3}{2}}} \, ds ∫02​(16−s2)23​3s+4​ds
[3]

1.11 H: Integration Questions

  1. A Level
  2. /Maths
  3. /1.11 H: Integration

Practise AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions 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), matched to the AQA A Level Maths (7357) 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.

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