page 04 MARKS 6. A body moves with an initial velocity u m s−1 in a straight line on a smooth horizontal surface. It travels with a constant acceleration of a m s−2. After t seconds it has a velocity v m s−1. (a) Use calculus to show that v u at   . (b) Hence, derive an expression for its displacement, s metres, from its original position in terms of u, a and t. 7. Use integration by parts to find sin 18 3 x x dx  . 8. A small object of mass m kilograms is placed on a rough horizontal disc, at a distance of 0.07 metres from the centre. When the disc is rotated at a rate of 1.5 revolutions per second, the object is on the point of slipping. Calculate the coefficient of friction between the disc and the object. 9. An object is formed by rotating the curve 3 6 3 1 x y x   between 1 x  and 2 x , through π 2 radians about the x‑axis. y O 1 2 x Using the substitution 3 3 1 u x  , or otherwise, find the exact value of the volume of this object. 2 2 3 5 6