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-axi