2022 Chp 6B Applications of Integration APQ Solutions (RVHS)

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6B Applications of Integration (Solutions to Tutorial and Assignment) 18 Additional Practice Questions (Chapter 6B Applications of Integration) 1. [2008/Promo/AJC/Q7] a) The diagram below shows (not to scale) the region R which is bounded by the curve 2 3 1 4y x  , the line 5 1y x  and the y-axis. Find the exact area of R. [4] b) Another region S is bounded by the curve 2 3 1 4y x  , the lines 3y x , x = 1 and the x- axis. Find the volume generated when S is rotated through 2 about the y-axis, giving your answer to 3 significant figures. [5] Solution: a.      0.5 20 1 2 21 0 1 3Area= 5 11 4 3 5tan 22 2 3 5 1 1 3 1 1tan 1 3 12 2 4 2 2 4 8 8 x dxx xx x                                  b. V1 = volume of cylinder =  2 31 5     = 3 5  V2 = 3 3 22 23 3 5 5 1 3 14x dy dy y         = 1.45210 V3 = volume of cone = 21 1 3 3 2 2 8           Required volume = V1+ V2-V3 = 2.94 x y R x = 1 y = 3x x y A B S Point A = (0.5, 1.5) from GC Point B, y = 3/5 Concept/Point of View For finding volume generated for part (b), it would be very tedious to use f ( ) g( ) d q p y y y approach. Instead, we can use consider “big overall volume minus unwanted volume” approach. We first determine V1 (vol of cylinder) and V2 and then subtract V3 (vol of cone) from the sum of V1 and V2!

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6B Applications of Integration (Solutions to Tutorial and Assignment) 19 2. [2008/Promo/ACJC/Q11] The diagram shows a sketch of part of the graph of 62y x x  . By considering the shaded rectangle and the area of the region between the graph and the x-axis for 2 3x  , show that 3 2 62 7x dxx     . [1] Show also that 3 2 62 8x dxx     . [1] Hence deduce that 1 1ln1.5p q  , where p and q are positive integers to be determined. [4] Solution: x y 0 2 3

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6B Applications of Integration (Solutions to Tutorial and Assignment) 20 3. [2008/Promo/HCI/Q13] i) Use the substitution 2 1u x  to find 2 1 dx x x . [3] The region R is bounded by the curve 2 1y x x  , the y-axis and line 1 2y . ii) Find the exact area of region R using your result in part (i). [3] iii) Find the volume of the solid generated when R is rotated through four right angles about the x–axis. [3] Solution: i 2 1 2 duu x dx    3 12 2 5 32 2 5 32 2 5 32 2 1 1 12 1 d d d2 2 4 1 1 4 4 1 1 (2 1) (2 1)10 6 ux x x u u u u u u u C x x C                ii Let 12 1 2x x  . From GC, x = 1 2 Area of R 1/2 0 1 1 2 1 d2 2 x x x      1/25 32 2 0 2 1 2 12 4 10 6 x x        2 2 2 2 2 1 1( )4 5 3 10 6 11 2 1 1 (11 2 4) units60 15 60             iii Volume of solid   2 12 2 0 1 1 2 1 d22 x x x              1/24 3 0 3 4 2 3 17 = 0.556 units96 x x          x y

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6B Applications of Integration (Solutions to Tutorial and Assignment) 21 4. [NJC/2009Promo/Q11] A curve C is defined by the parametric equations , where (i) Sketch C for 3,t showing the exact coordinates of the point of intersection of C with the x-axis. [2] (ii) The finite region S is bounded by C, the line ln 6x and the x-axis. Show that the area of S can be expressed as 29dqpttt, where p and q are constants to be determined. By using the substitution 3sect, find the exact value of this area. [7] Solution: 4(i) (ii) d 1 d x t t Area of S = ln 6 ln3 dy x = 6 2 3 19 dt t t      = 26 3 9 dt tt  (shown) Given d3sec 3sec tand tt      Area of S =   π 23 0 19sec 9 3sec tan d3sec         = π 23 03 tan d  = π 23 03 sec 1 d  =   π 303 tan  = π3 3 3     = 3 3 π ln6 x O y (ln3, 0)

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6B Applications of Integration (Solutions to Tutorial and Assignment) 22 5. [2009/Promo/NJC/Q10(b)] The diagram above shows part of the graph of a curve C given by 1 yx y . The region R is bounded by C and the lines y x and 4x. Write down the equation of the curve obtained when C is translated by 4 units in the negative x-direction. Hence, or otherwise, write down also an expression for the volume V of the solid formed when R is rotated 2π radians about the line 4x and find its numerical value, giving your answer correct to 3 decimal places. [5] R 4 x y x O y 1 yx y  1 yx y  x Solution: 4 4 1 1 y yx x y y           22 2 4 3 3 Volume 4 d 2 21 3 8ππ 1.40833 (by GC)3 12.80197 12.802 units (to 3 dec. pl.) y yy            

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6B Applications of Integration (Solutions to Tutorial and Assignment) 23 6. [RVHS/2009/Promo/Q12] The region A is bounded by the curves 1 tan2y x and siny x(see diagram). (i) Verify that the x-coordinate of the point of intersection of the 2 curves, P, is .4  [1] (ii) Find the exact area of the region A, giving your answer in the form ln 2a b where a and b are exact constants to be determined. [4] (iii) Find the exact volume of the solid of revolution formed when the region A is rotated through 360about the x-axis. [4] (iv) The region A is rotated through 360about the y-axis instead. Find the volume of the solid of revolution in this case, giving your answer correct to 4 decimal places. [2] 1 tan2y x siny x A O P x y Solution: (i) Since 1 1 1sin , tan4 4 2 2 2    , x-coordinate of P is .4  (ii) Area of A 4 4 0 0 1sin d tan d2x x x x      4 40 0 1 sin[cos ] d cos2 xx x x      40 1 11 [ln cos ]2 2 x     1 1 11 ln( )2 2 2   2 21 ln 2.2 4   (iii) 2 24 4 0 0 1sin d tan d2xV x x x x       24 4 0 0 1 cos 2 d tan d2 2 x x x x       24 40 0 1[ sin 2 ] sec 1 d2 2 2x x x x        40 1[ ] [tan ]2 4 2 2 x x       2 3 .4 4    (iv) 1 1 1 2 1 22 2 0 0(tan 2 ) (sin )yV y dy y dy      0.1280.

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6B Applications of Integration (Solutions to Tutorial and Assignment) 24 7. 2015 NJC Promo/ 6