Additional Mathematics Chapter 10: Calculus II - Integration Chapter Assessment 1 2 (i) Given that (ii) Find dy 4 x3 6 x 2 5 , find y. dx [3] 2 x 3 dx . [3] A curve passes through the point (1, 1) and has gradient function dy 4x 5 . dx Find the equation of the curve. 3 [4] Find the value of the following. 3 (i) 4 x 1 dx [3] 0 1 (ii) 3x 2 x 1 dx [4] 2 4 Find the area of the shaded region between the curve y x 2 x 7 , the line x = 3 and the axes. [4] © MEI, 18/04/05 1/3 Additional Mathematics 5 The solid side of a small rocking toy has the shape given in Fig. 5. The curve ABC has equation y x 2 4 x 4 , D has coordinates (6, 16), the side DE has equation 2y = 3x + 14 and the side EA is the y-axis. D E C A B Fig.5 Find the area of the shaded region representing the side of the rocking toy. Units are centimetres. 6 (i) Calculate the coordinates of the points A and B where the curve y = 6x x2 meets the straight line y = x + 4. [4] (ii) Sketch the curve and the line and shade the area that is completely enclosed by the curve and the line. (iii) Calculate this shaded area. 7 [7] [3] [5] A curve has equation y = x3(4 x). (i) Find the values of x for which y = 0. [2] (ii) Find the coordinates of the turning points. [5] (iii) Calculate the area enclosed by the curve and the x-axis. © MEI, 18/04/05 [3] 2/3 Additional Mathematics 8 2 roads, AB and CD, meet at a junction. AB runs due East-West and CD runs North-South. At the junction there is an arm from A that turns northwards to meet the other road at C and an arm from A that turns southwards to meet the other road at D. The shape enclosed is a grassed area. The plan of this junction may be modelled as follows: Road CD is the y-axis and road AB is parallel to the x-axis. D is the origin. The vertices of the grassed area are A, C and D. 1 The equation of the upper arm from A to D is y x 2 20 x 200 . 5 1 The equation of the lower arm from A to D is y x 20 x . 5 A is at the point which is a maximum for one curve and a minimum for the other curve. C A B D (i) Find the coordinates of the points A and C. [5] (ii) Find the area of the grass. [5] Total: 60 © MEI, 18/04/05 3/3