DHS Applications of Integration (9649) Topical Revision

Applications of Integration H2 Double Math Applications of Integration [part FM] 1 (a) (i) The shaded region R in the diagram above is bounded by th e curve with equation 1tan ,yx  the line with equation π 114yx  and the y-axis. Show that the exact area of R is 14 πln 228  . [5] (ii) Find the volume generated when R is rotated completely about the y-axis. [3] (b) A curve has parametric equations 2ln , sin ,x tt y t where 1 π.t Find the numerical value of the area of the region bounded by the curve, the x-axis and the y-axis. [3] [2014 JJC/Promo/9] 2 (a) Find edxxx  . [3] (b) The curve C has parametric equations 2(2 ) , e , for 2.2 ttxy t    (i) Show that the equation of the normal to the curve at t = 0 is 23yx . [4] (ii) Find the exact value of the area of the region bounded by C, the line 23yx , the x-axis and the y-axis. [4] [ 2014 RVHS/Promo/11] O 1 R

Applications of Integration 3 (a) Using an algebraic method, find the exact value of 4 1 2 dx xx  . [3] (b) Sketch and shade the finite region bounded by the curve y = x2 + 2, the lines y = x and x = 1, and the y-axis. Find the exact volume of the so lid formed when the region is rotated 2  radians about the y-axis. [4] [2014 TJC/Promo/6] 4 (i) Use the substitution 2t a nx  to show that 1 22 2 11 2 dt a n16 2(4 ) 4 xxxC xx    . [5] (ii) The region R is bounded by the curve 1 4y x   , the line 2y , the x-axis and the y-axis. Find the exact volume of the solid formed when R is rotated 2π radians about the y- axis. [4] [2014 RVHS/Promo/10] 5 (a) Find 2( 62 d14 ) x xxx  [5] (b) The diagram above shows the region R bounded by the curve C with equation ()ln ,1 ,xyx x  the x-axis and the line L with equation 1 (2 ) .e( e 2 )yx  Find the exact volume of the solid of revolution when R is rotated completely about the x-axis. [5] [2014 AJC/1/3] L C R e x y

Applications of Integration 6 The graph of (4 0 )xy  is shown in the diagram below. By considering the shaded rectangle, show that 1 40 40 d . n n nx x     [1] Deduce that 41 40 1 2 3 . . . 80 40 d . xx      [2] Show also that 1 41 40 d . n n nx x     [2] Deduce that 41 40 1 2 3 . . . 81 40 d . xx      [1] Hence, deduce the value a , where a  , that satisfies the following inequality, 9 1 2 3 ... 8 0 9 1 .aa  [2] [2014 AJC/1/9] 7 A curve has equation given by  2 ln 1yx , where 0x . (i) Sketch the graph, indicating the exact coordinates of the x -intercepts and the turning point. [4] The region R is bounded by the curve and the -axis. (ii) Find the exact