Skip to content
MathsGenie logo
Open app

Course home

  1. A Level
  3. Maths AQA
  5. Question bank

1.11 H: Integration

EasyMediumHard
123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355356357358359360361362
Question 167
a.

Using the substitution u=3x+32sin⁡2xu = 3x + \frac{3}{2}\sin 2xu=3x+23​sin2x, show that

∫0π2e3x+32sin⁡2xcos⁡2x dx=16(e3π2−1) \int_0^{\frac{\pi}{2}} e^{3x + \frac{3}{2}\sin 2x} \cos^2 x \, dx = \frac{1}{6}(e^{\frac{3\pi}{2}} - 1) ∫02π​​e3x+23​sin2xcos2xdx=61​(e23π​−1)
[4]
b.

The design of a high-performance aerodynamic component involves a surface generated by rotating a region RRR through 2π2\pi2π radians about the xxx-axis. The region RRR is bounded by the curve with equation

y=18e32x+34sin⁡2xcos⁡x y = \sqrt{18} e^{\frac{3}{2}x + \frac{3}{4}\sin 2x} \cos x y=18​e23​x+43​sin2xcosx

and the coordinate axes in the first quadrant.

Use the result from part (a) to find the volume of the component, giving your answer in simplest form.

[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.

Question bank