(i) Find the exact value of p. [4] Show that the area of the shaded region bounded by the curve, the x-axis and the line

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1 H Math : Integration Apps 0. M p The diagram shows the curve e e and its maimum point M. The -coordinate of M is denoted b p. (i) Find the eact value of p. [] (ii) Show that the area of the shaded region bounded b the curve, the -ais and the line p is equal to. [] 8. Q The diagram shows the curve the curve. sin, for 0. The point Q, lies on (i) Show that the normal to the curve at Q passes through the point,0. [5] d d (ii) Find sin cos. [] (iii) Hence evaluate sin d. [] 0 KL Ang Jan 0 Page 67

2 H Math : Integration Apps 0. M P The diagram shows the curve e. (i) Show that the area of the shaded region bounded b the curve, the -ais and the 7 line is equal to. [5] e (ii) Find the -coordinate of the maimum point M on the curve. [] (iii) Find the -coordinate of the point P at which the tangent to the curve passes through the origin. []. M The diagram shows the curve M. 5sin cos for 0, and its maimum point (i) Find the -coordinate of M. [5] (ii) Using the substitution u cos, find b integration the area of the shaded region bounded b the curve and the -ais. [5] Page 68

3 H Math : Integration Apps 0 5. C A R B The diagram above shows the curve C with parametric equations 5 t, t9 t The curve C cuts the -ais at points A and B. (i) Find the -coordinates of points A and B. [] The shaded region, R, is enclosed b the loop of the curve C. (ii) B integration, find the area of R.. [6] 6. a b The curve shown in the diagram above has an equation. A shaded region is bounded b the curve, -ais and the lines a and b as shown in the diagram. The region is rotated 60 about the -ais to generate a solid of revolution. Find the volume of the solid generated. Epress our answer as a single fraction, in terms of a and b. [5] KL Ang Jan 0 Page 69

4 H Math : Integration Apps 0 7. (a) Using the substitution cos u or otherwise, find the eact value of d. [7] The diagram above shows a sketch of part of the curve of, 0. The shaded region is bounded b the curve, -ais and the lines and as shown in the diagram. This region is rotated radians about the -ais to form a solid of revolution. (b) Using the result in part (a), find the eact volume of the solid of revolution form. [] 8. (a) is the origin and A is the point on the curve tan where. Show that the area of the region enclosed b the chord A and the arc A of the curve is 8 ln. [6] (b) A portion of the curve a, where a is a positive constant, is rotated about the vertical ais to form the curved surface of an open bowl. The bowl has a horizontal circular base of radius r and a horizontal circular rim of radius r. 8r Prove that the depth of the bowl is. [] a Find the volume of the bowl in terms of r and a. [] a Given that the volume of the bowl is 0, find the depth of the bowl in terms of a onl. [] Page 70

5 H Math : Integration Apps 0 9. (a) Using the identit cos sin, find sin d. [] S C The diagram above shows part of the curve C with parametric equations tan, sin 0 The finite shaded region S is bounded b C, the line and the -ais. This region is rotated radians about the -ais to form a solid of revolution. (b) Show that the volume of the solid of revolution formed is given b the integral k 6 0 sin d where k is a constant. [5] (c) Hence find the eact value of this volume, giving our answer in the form p q, where p and q are constants. [] 0. (i) Find, correct to significant figures, the coordinates of the turning point of the curve sin for which 0. Hence sketch the curve for 0. (ii) The region bounded b the curve sin, the -ais and the line is rotated through radians about the -ais. Find the volume of the solid of revolution so formed, giving our answer in terms of. [] [] KL Ang Jan 0 Page 7

6 H Math : Integration Apps 0. C R ln ln The curve C with parametric equations lnt, t t The finite shaded region R is bounded b C, the lines ln and ln, and the -ais. (a) Show that the area of R is given b the integral 0 d t t t. [] (b) Hence find the eact value for this area. [6] (c) Find a cartesian equation of the curve C, in the form f. [] (d) State the domain of the values for for this curve. []. Given that z, show that dz d. [] Find the eact value of the area of the region bounded b the curve -ais and the lines and 7. [], the Page 7

7 H Math : Integration Apps 0. The diagram above shows a sketch of part of the curve of,. The shaded region is bounded b the curve, -ais and the lines and as shown in the diagram. This region is rotated 60 about the -ais to form a solid of revolution. (a) Use calculus to find the eact value of the volume of the solid generated. [5] A B The bell-shaped figure above is a paperweight with ais of smmetr AB where AB cm. A is a point on the top of the paperweight, where B is a point on the base of the paperweight. The paper weight is geometricall similar to the solid in part (a). (b) Find the volume of this paperweight. []. Given that e, prove b induction that, for all positive integers n, n e n d n nn nn n. d [5] Hence find e 68 6d. [] KL Ang Jan 0 Page 7

8 H Math : Integration Apps 0 5. P sin A A cos The region bounded b the aes and the curve divided into two parts, of areas A and A, b the curve cos from 0 to sin. is Prove that A A. [6] The two curves meet at P. The line through P parallel to the -ais meets the -ais at Q. The region PQ, bounded b the arc P and the lines PQ and Q, is rotated through right angles about the -ais to form a solid of revolution of volume V. It is given that V 0 sin d. (i) B substituting u sin, show that V u cos u du. [] 0 d (ii) Show that u sin u u cos u sin u u cos u du. [] (iii) Hence find the eact value of V. [] Page 7

9 H Math : Integration Apps 0 Answer kes:.. (i) ln (ii) sin (iii).. (ii) (i) (iii) (ii) (i), cos 8-5 b a a b (ii) (a) (b) (b) 0 r a ; a (a) sin c (i).9rad (b) k 6 (c) (ii) 8.. (b) ln 7 (c) e (d) 0 KL Ang Jan 0 Page 75

10 H Math : Integration Apps 0.. (a) 6 (b) 5. e 6 0 c Page 76