DHS Y4 Math 2 Supplementary Worksheet Integration(Area of Region)

1 Year 4 Mathematics 2 Applications of Integration Area of a Region Supplementary Worksheet Name : _________________________________ ( ) Class : _______ Date : __________ 1 (a) Evaluate 0.6 2 0.1 4cos 2 dxx , giving your answer correct to 2 decimal places. (b) The diagram shows part of the graphs of 12 += xy and xy 2cos= . Find (i) the coordinates of A, B and C, (ii) the area of the shaded region giving your answer correct to 2 decimal places. [2008 CHIJ Toa Payoh Sec AMaths P2] 2(a) The diagram shows part of the curve g( )yx= . Copy this diagram and use it to explain why 3 1 4 g( ) d 6xx . (b) Given that 3 1 f ( ) d 5xx = , evaluate 13 22 1 f ( ) d f ( ) dx x x xx  −+ . [2008 Commonwealth Sec AMaths P1] O x y C B xy 2cos= 2 1yx=+ A y x 1 3 4 (3, 4) g( )yx= 2 (1, 2) • O •

2 3 The diagram shows part of the graph of 2e xy= and of 21 secyx=− . OABC is a rectangle where A is on the y-axis, B is on the curve 2e xy= and C is (ln 2, 0). (i) Find the area of the shaded region, which is bounded by the two curves, the y-axis and the line ln 2x= . (ii) Region P is bounded by the curve 2e xy= , the line AB and the y-axis. Given that the area of region P may be expressed as 1 f ( ) d k yy , find the value of k and function f(y). [2008 Commonwealth Sec AMaths P2 (modified)] 4(i) The diagram below shows part of the curve xy 2sin= and the line 2 1=y . Find the area of the shaded region. A O B C (ln 2, 0) x y 2e xy= 21 secyx=− P y x O 2 1=y xy 2sin=

3 (ii) The diagram below shows part of the graph 2ln xy = , cutting the x-axis at (1, 0). The line y = 2 intersects the curve at P. A line is drawn from P, parallel to the y-axis, to meet the x-axis at Q. (a) Find the x-coordinate of Q. (b) Differentiate xx ln2 with respect to x. (c) Hence find the area of the shaded region. [2008 Crescent Girls’ AMaths P2 (modified)] 5 The diagram above shows part of the curve xxy 4+= passing through the points P, Q and R. (i) The curve has a minimum point at P. Find the coordinates of P. (ii) Given that the gradient of the line PQ is −1, find the coordinates of Q. (iii) Calculate the area of the shaded region. [2008 Holy Innocents High AMaths P2] O 1 2ln xy = Q P 2 x y y x P Q 8 0 R

4 6 The diagram shows part of the curve 2222 xxy = + + and the straight line BC is the normal to the curve at the point C(−2, 1). (i) Find the equation of the line BC. (ii) Show that 82 5OA BC= . (iii) Find the area of the shaded region. [2008 Singapore Chinese Girls’ AMaths P2] 7(a) Given that 12 += xxy , show that 12 45 d d 2 + += x xx x y . Hence, evaluate 2 3 0 15 12 d 1 xx x x + + . (b) The diagram shows part of the curve xy 2sin= . The line OA intersects the curve at A where the y-coordinate is 2 3 . (i) Find the x-coordinate of A in terms of . [1] (ii) Find the area of the shaded region. [4] [2008 Tanjong Katong Sec AMaths P2] O x y A B C(−2, 1) 2222 xxy = + + y = sin 2x x y O A

5 8 In the diagram, the curve 2 23yx=− and the straight line 3xy+= intersect at two points A and B. Find (a) the coordinates of A and of B, (b) the area of the shaded region. [2008 Temasek Sec AMaths P2] 9 The figure below shows parts of the curve 242 xy −= and 2)2(2 += yx . Calculate (i) the coordinates of A, B and C, (ii) the shaded area enclosed by the curves and the y-axis. [2008 Unity Sec AMaths P2] 2 23yx=− 3xy+= O A B y x y x B A O C 2)2(2 += yx 242 xy −=

6 10 The diagram shows part of the curve of 11 223e e xx y − =+ . (i) Show that the exact value of the y -coordinate of the stationary point of the curve is 32 . (ii) Calculate the area enclosed by the curve, the x-axis and the lines x = 0 and x = 1. [2008 Zhonghua Sec AMaths P1 (modified)] 11 The diagram shows part of the curve y x= −10 32 2 and two parallel lines OR and PQ. The line OR intersects the curve at the point (2,2)R and the line PQ is a tangent to the curve at the point Q. Find (a) the gradient of OR, (b) the coordinates of Q, (c) the area of the shaded region OPQR. [2008 Zhonghua Sec AMaths P2 (modified)] O x y 11 223e e xx y − =+ 4 P O (2,2)R Q y x y = x y x= −10 32 2

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8 Answers 1(a) 1.14 (b)(i) ( ) ( ) ( ) 2πππ ,0 4 1640 ,1 , , , 1A B C + (ii) 0.45 unit2 2(b) 4.31 3(i) 1.64 units2 (ii) k = 4 ; ( ) 1 2f lnyy= 4(i) 0.342 units2 (ii)(a) e (b) ( )2 ln 1x+ (c) 2 units2 5(i) (2 , 4) (ii) (1 , 5) (iii) 42.3 units2 6(i) 3y + 4x + 5 = 0 (iii) 3.35 units2 7(a) 108 (b)(i) 1 6 π (ii) 0.0232 units2 8(a) A(2, 1) ; B(6, −3) (b) 1 35 units2 9(i) A(0, 2) ; B(2, 0) ; C(0, −2) (ii) 4 units2 10(ii) 3.66 units2 11(a) 1 (b) (4, 8) (c) 10 units2