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 area of R . [4] (iii) Find the volume of the solid generated when R is rotated through 2π radians about the y -axis. [4] [2014 HCI/1/11] x 0 41 n n+1 x y

Applications of Integration 8 A curve C is defined by the parametric equations cos , sin 2 , for 0 2 π.xt y t t   (i) Sketch the curve, stating the coordinates of any points of intersection with the axes. [2] (ii) Show that the area enclosed by the curve C is π 22 0 8s i nc o s d .tt t Hence, find the exact area enclosed by the curve C. [4] (iii) Find the exact value of the volume of revolution formed when the area enclosed by the curve C is rotated completely about the x- axis. [3] [2014 MI/I/8(modified)] 9 The curve C has equation 13 22 1 3 xyx  where 0  and 03 x . The length of C is denoted by s. Prove that 23s . [4] The area of the surface generated when C is rotated through one revolution about the x-axis is denoted by S. Find S in terms of . [5] [FM/N06/I/12ORpart] 10 Prove that 2 2 d1 1 1ln , 0.d 1 x xxx xx         Hence, or otherwise, find the exact le ngth of arc of the curve whose equation is lnyx from the point where 1 3 x to the point where 1.x [SRJC2001/1/4] 11 The parametric equations of a curve C are given by 1 2tan , ln(sec ) for 0 π.xt t y t t    (i) On the curve C, O and P are points corresponding to 0t  and tp respectively. Prove that the arc length of OP of C is sec 1.p [3] (ii) The arc OP of length 1 unit is rotated about the x-axis through 4 right angles. Find the area of the curved surface so formed, leaving your answer in exact form. [5] [TJC FM2001/1/4 (modified)]

Applications of Integration 12 A curve is defined parametrically by e cos , e sin where 0 and ttx at y a t t a  is a positive constant. (i) Show that 1 4 d tan( ).d πy tx  [3] (ii) A and B are points on this curve corresponding to 0t  and π.t  Find the length of the arc AB. [4] (iii) The portion of the curve from 0t  and πt  is rotated through 2π radians about the x-axis. Prove that the area of the curved surface generated is 22 π22 π (1 e )5 a  units2. [5] [FMSAJC 2001/1/5] 13 A curve has the parametric equations 1 2(cos 1), (sin ), where 0 t πxa t ya tt    and a is a positive constant. The arc of the curve between 0t  and 1 3 πt is rotated through 1 revolution about the x-axis. Show that the area of the surface of revolution formed is given by π 2 3 0 4π (sin )sin d . 2 tA at t t   Hence, or otherwise, show that   22 π 23 π 11 .3Aa [8] [TPJC FM 2001/1/5] 14(a) Let 22 0 Is i n d n n x x   , n   . (i) Show that 1 21II 2 nn n n   , n   . [3] (ii) Find 0I and 5I in terms of  . [3] (b) By using the shell method, find the exact volume of the solid of revolution formed by rotating the region bounded by the graphs of 1yx and 2(1 )yx about the y-axis. [3] [2016 HCI/Promo/8]

Applications of Integration 15 3D printing is a term that is commonly used to describe any process in which a 3D object is created from a computer model. To be able to print in 3 dimensions, a computer-aided manufacturing (CAM) software is used to control the motors a nd the movement of a 3D printer nozzle which is the part that does the actual printing. A certain CAM software is designed to move the nozzle along the path C described by the parametric equations 33 sin , cos 2xy   for 1 20 π . Find 2 2 d d y x in terms of . [4] Find the exact length of the path covered by the nozzle, from 0  to 1 2 π  . [3] After completion of the printing for C, the printer continues to print on the surface generated by rotating C through 2π radians about the x-axis. Find the exact area of the surface generated. [4] [2016 HCI/Promo/10] 16 A newly discovered planet revolvi ng around a distant star is modelled to be in an elliptical trajectory described by the polar equation below, with the star positioned at the origin and x-axis as the major axis. 1 10 . 3 c o sr   (i) Show that the distance covered by the planet as it moves from the point A where θ = 0 to the point B where θ = 1 2 π is given by  π 2 20 1 1.09 0.6cos d 10 . 3 c o s D     . [3] (ii) Using trapezium rule with 5 ordinate s, approximate the integral in part (i) to 4 decimal places. [2] (iii) Using Simpson’s