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1.11 H: Integration

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Question 121
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]
Markscheme

1.11 H: Integration Questions

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

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.

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