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Question 559

A glass component for a specialized microscope objective is designed as a solid of revolution. The component's profile is created by rotating a specific curve through 360∘ 360^\circ\,360∘ radians about the xxx-axis, where the units are centimetres.

The profile of the glass component is modeled by the equation

h(x)=13(2−x)ex,0≤x≤2 h(x) = \frac{1}{3}(2 - x)e^x, \quad 0 \le x \le 2 h(x)=31​(2−x)ex,0≤x≤2
a.

Show that the volume, V cm3V \text{ cm}^3V cm3, of the silica glass component is given by

V=K∫02(x2−4x+4)e2x dx V = K \int_{0}^{2} (x^2 - 4x + 4)e^{2x} \, dx V=K∫02​(x2−4x+4)e2xdx

where KKK is a constant to be determined.

[2]
b.

Hence, calculate the exact volume of the glass component. Give your answer in the form pπ(eq+r) cm3p\pi(e^q + r) \text{ cm}^3pπ(eq+r) cm3, where p,q,p, q,p,q, and rrr are rational numbers to be found.

[5]
Markscheme

Integration Questions

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

864 exam-style questions on Edexcel A Level Maths Integration, covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations. Each one has a worked solution and a mark scheme showing where the marks go.

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