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1EDEXCEL C4 June 2009 Q8
(a) Using the identity cos 2theta = 1 - 2 sin^2 theta, find the integral of sin^2 theta dtheta. (2) Figure 4 shows part of the curve C with parametric equations x = tan theta, y = 2 sin 2theta, 0 <= theta < pi/2. The finite shaded region S shown in Figure 4 is bounded by C, the line x = 1/sqrt(3) and the x-axis. This shaded region is rotated through 2pi radians about the x-axis to form a solid of revolution. (b) Show that the volume of the solid of revolution formed is given by the integral k times the integral from 0 to pi/6 of sin^2 theta dtheta, where k is a constant. (5) (c) Hence find the exact value for this volume, giving your answer in the form p pi^2 + q pi sqrt(3), where p and q are constants. (3)
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2EDEXCEL C4 January 2008 Q4
(i) Find the integral of ln(x/2) dx. (4) (ii) Find the exact value of the integral from pi/4 to pi/2 of sin^2 x dx. (5)
