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1.8 Integration

1.8 Integration

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Question 166
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

1.8 Integration Questions

  1. A Level
  2. /Maths
  3. /1.8 Integration

438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.

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