SPM Past Year Questions : AM Form 5 Chapter 3 - Integration
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1 SPM Past Year Questions : AM Form Chapter - Integration SPM PAST YEAR QUESTIONS ADDITIONAL MATHEMATICS FORM CHAPTER : INTEGRATION SPM 99 a) Find i) (4 - )(4 ) d, ii) 8 ( - ) d b) Given f ( ) d 8, find the value for f ( ) d. ( marks) a) Calculate the volume of the solid generated is rotated through 6 about the y - ais. when the shaded region in the diagram ( marks) b) The diagram shows the front surface of a door. AB DCare sting vertically BC has thesame shapewith part of the graph y Given AB DC m K is the highest point from the level of BC. (i) Calculate the width of BC. is lying horizontally. AKD is a symmetrical arc that (ii) Calculate the area of the door's front surface. (iii) If the height of AB DC are increased to 4m, whereas the width of BC theshapeof the arc the front surface of the new door. AKD are maintained, find the area of (7 marks) Compiled by Miss Page
2 SPM Past Year Questions : AM Form Chapter - Integration SPM 994. Find ( 7) d. ( marks) 4. Given dy d 4, find y in terms of if y 6 when. ( marks). The diagram shows the graphs of y 4 7 y 7 -. Calculate the area of the shaded region. 6.a) The diagram shows part of tangent to the curve at the maimum point,r. Calculate the volume of solid generated when the shaded region is revolved through 6 about the - ais. the graph y 6 -. PR is a b) The gradien function of the two tangents to the twocurves are given by respectively.if the curves intersect at point (,4), find the equation of each curve. SPM 99 7.a) Evaluate ( - )( ) d. ( marks) b) Given for - - y g()d. dy g() d where g() is a function in. Find the value ( marks) Compiled by Miss Page
3 SPM Past Year Questions : AM Form Chapter - Integration 8.a) The diagram shows part of at point Q. shaded region. thecurve y Given that the straight line PQ 4( ) that intersects with thestraight line is parallel with - ais. Calculate the area of the b) In thediagram, KMNis part of thecurve which is symmetrical at y - ais. Given OKMNis a sector of a circle with thecentre at O KON 9.Calculate the volume generated when thesector OKMN is revolved through angle 8 at y - ais. y (6 marks) SPM Given a) 4 4 f()d f()d g()d, g()d 6. Find the value of ( marks) b) k if g() - kd 4. ( marks) Compiled by Miss Page
4 SPM Past Year Questions : AM Form Chapter - Integration.a) The diagram shows the curve by thecurve, - ais, y ( -)( ). line - line. Calculate the area of the region bounded (6 marks) b) The diagram shows the shaded region bounded by thecurve line 8 unit k. When the region is. Find the value of k. y, line rotated 6 about the - ais, the volume generated is SPM 997.Given f()d 6, find the value for f() - d. ( marks).a) The diagram shows a straight line at point (h,). Find (i) the value of h, (ii) the area of shaded region. y a curve y ( - )( - 4) which intersect Compiled by Miss Page 4
5 SPM Past Year Questions : AM Form Chapter - Integration b) The diagram shows theshaded region bounded by shaded region is revolved 6 about y - ais, the volume generated is 6 unit. the value of k. y 4 -, y k y - ais. When the Determine SPM 998.a) Find. 4 d b) Find the equation of the curve that has a gradient of ( ) ( marks) through point,. (marks) 4.a) Sketch the graph y 4 in the range of 6. Hence find the area of the region bounded by y 4, - ais line 6. y b) The diagram shows the graph y straight line. Find the volume generated when the shaded region is revolved through 6 about y - ais. Compiled by Miss Page
6 SPM Past Year Questions : AM Form Chapter - Integration SPM 999 d y.given 4 d. When -, y dy d. Find y in terms of. 6.The diagram shows the graph y f(), which touches - ais at point A cuts y - ais at B. The straight line BC, which is parallel with - ais, Given f' () a) Find (i) the coordinate of point A, (ii) f(), (iii) the coordinate of Point B. Hence, calculate 4. the area of the shaded region. is the tangent to the curve at point B. ( mark) ( marks) ( mark) SPM 7.Given a) f()d, f()d 4. b) k constant if Find the value of f()d f()d k d. ( mark) 8. With reference a) Calculate b) Q is a block which is produced when the region bounded by thecurve y line y a is revolved through 8 about y - ais. If the volume produced is find the value of a. to the diagram, answer the following questions. the area of shaded region. unit, (6 marks) Compiled by Miss Page 6
7 SPM Past Year Questions : AM Form Chapter - Integration SPM 9.a) Find the value of b) The diagram shows a sketch of (i) Shade (ii) Hence, find the region represented by the value of 6d. part of 8 yd dy. 8 a curve graph. dy. ( marks) ( marks).given d d g( ). Find the value for - g() d. ( marks).a) Given (i) Find the value of (ii) The diagram shows a straight line y g() at point (,9). - ais g()d 4 where g() is a linear function. Find the area bounded by thestraight line y g(), the curve y ( - ), y - ais. g()d. which intersects the curve y ( - ) (6 marks) Compiled by Miss Page 7
8 SPM Past Year Questions : AM Form Chapter - Integration b) The diagram shows a shaded region bounded by part of AB of for point y - ais. It is known that the straight line A is k - revolved 6 about the y - ais is 8 unit AB is parallel with - ais where k is a constant.if the volume generated when the region is, find the value of k. the curve y, the straight line thecoordinate SPM.On a specific day,the rate of increase in the temperature,, compared to the time, t s, is d given by dt t. ( t). Given that 4 when t 6, find the maimum value of when ( marks).a) Given dy 4. d Epress y in terms of. ( marks) b) Given 6 8k k d 7, where k is a constant.find the possible values of k. ( marks) Compiled by Miss Page 8
9 SPM Past Year Questions : AM Form Chapter - Integration 4.a) The diagram shows part of the curve y. The straight line t divides theshaded region bounded by thecurve y, the straight line, - ais into two regions, P Q. Given that thearea of region P is three times the area of region Q, calculate the value of t. b) The diagram shows the graph of shaded region bounded by thecurve, straight line straight line y 6 is revolved through 6 about y - ais. a curve y.the - Find the volume of the block which is generated. SPM. d k( ) n c, 4 ( ) find the value of k the value of n. ( marks) 6.The diagram shows a curve y a straight line k. If the area of the shaded region is 64 units, find the value of k. ( marks) dy 7.Given y 6 when -, find d y in terms of. ( marks) Compiled by Miss Page 9
10 SPM Past Year Questions : AM Form Chapter - Integration 8.The diagram shows a curve y straight line y at point A. Calculate the volume generated when the shaded region is revolved through 6 about y - ais. which intersect a (6 marks) SPM 4 9.Given k - ( ) d 6, where k -, find the value of k.. The gradient function of a curve that passes through Find the equation of the curve. A(,-) is 6. ( marks).the diagram shows part of a curve y ( -) passesthrough point A(,).A region is bounded by the curve, - ais, straight line, straight line which. (a) Find the area of the region. (b) The region is revolved through 6 about - ais. Find the volume generated, in terms of. (6 marks) SPM.Given 6 f ( ) d 7 6 f() - k d. Find the value of k.. A curve has a gradien function of 4 through point (,). Find the equation of the curve. ( marks) Compiled by Miss Page
11 SPM Past Year Questions : AM Form Chapter - Integration 4. In thediagram, straight line PQis a normal to the curve y at A(,). Find (a) The area of the shaded region. (b) the volume generated, in terms of, when the region bounded by thecurve, y - ais is revolved through 6 straight line y about the y - ais. ( marks) SPM 6.The diagram shows thecurve y f() cutting the - ais b a f()d. at a the shaded region is units 6.Given that (a) the value of (b) the value of b. Given that the area of g()d 8, find k if g()d,, find the value of ( marks) k - g() d. 7. The diagram shows the straight line y 4 intersecting the curve y ( - ) Find (a) the value of k, (b) the area (c) the volume generated, in terms of, when theshaded region of theshaded region P, at thepointsa B. Q is revolved 6 about the - ais. ( marks) ( marks) Compiled by Miss Page
12 SPM Past Year Questions : AM Form Chapter - Integration SPM 7 8.Given that (a) (b) 7 7 h()d, 7 - h() d h()d, find 9. A curve with gradient function - Find the equation of the curve., has a turning point at (,8). ( marks) 4.The given diagram shows part of the curve y ( -) The curve intersects the straight line at point A.. Calculate (a) the area of theshaded region P, (b) the volume generated, R which is bounded by thecurve, the - ais the y - ais in term of, when the region is revolved through 6 about the - ais. (7 marks) Compiled by Miss Page
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