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