(ii) Explain how the trapezium rule could be used to obtain a more accurate estimate of the area. [1]

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1 C Integration. June 00 qu. Use the trapezium rule, with strips each of width, to estimate the area of the region bounded by the curve y = 7 +, the -ais, and the lines = and = 0. Give your answer correct to significant figures. [4] Eplain how the trapezium rule could be used to obtain a more accurate estimate of the area. []. June 00 qu.6 (a) Use integration to find the eact area of the region enclosed by the curve y = + 4, the -ais and the lines = and = 5. [4] (b) Find ( 6 y ) dy. [] (c) Evaluate 8 d. [4]. Jan 00 qu. The gradient of a curve is given by points (, 5) and (p, 5). dy d = 6 4. The curve passes through the distinct Find the equation of the curve. [4] Find the value of p. [] 4. Jan 00 qu.4 Use the trapezium rule, with 4 strips each of width 0.5, to find an approimate value for 5 log 0 ( + )d, giving your answer correct to significant figures. [4] 5 Use your answer to part to deduce an approimate value for log0 + d, showing your method clearly. [] 5. Jan 00 qu.5 The diagram shows parts of the curves y = + and 9 y =, which intersect at (, ) and (, 0). Use integration to find the eact area of the shaded region enclosed between the two curves. [7]

2 6. June 009 qu.4 Find the binomial epansion of ( 5), simplifying the terms. [4] Hence find ( 5) d. [4] 7. June 009 qu.6 dy The gradient of a curve is given by d = + a, where a is a constant. The curve passes through the points (, ) and (, 7). Find the equation of the curve. [8] 8. June 009 qu.9 Sketch the graph of y = 4k, where k is a constant such that k >. State the coordinates of any points of intersection with the aes. [] The point P on the curve y = 4k has its y-coordinate equal to 0k. Show that the -coordinate of P may be written as + log k 5. [4] (iii) (a) Use the trapezium rule, with two strips each of width, to find an epression for 9. Jan 009 qu. the approimate value of 4 d. [] (b) Given that this approimate value is equal to 6, find the value of k. [] Find ( + 8 5)d, [] d. [] 0. Jan 009 qu.4 0 k The diagram shows the curve y = 4 + and the line y = 9 which intersect at (, 9) and (, 9). Use integration to find the eact area of the shaded region enclosed by the curve and the line. [7]

3 . June 008 qu.5 The diagram shows the curve y = + +. The shaded region is bounded by the curve, the y-ais, and two lines parallel to the -ais which meet the curve where = and = 4. Show that the area of the shaded region is given by y 6y + 7 dy [] Hence find the eact area of the shaded region. [4]. June 008 qu.7 (a) Find + 5 d. [4] (b) Find 8 4 d. []. June 008 qu.9 ( ) ( ). 4 Hence evaluate 8 d. [] 7 5 (b) Use the trapezium rule, with four strips each of width 0.5, to find an approimate value for cos d, where is in radians. Give your answer correct to significant figures. [4] 0 4. Jan 008 qu. 7 Use the trapezium rule, with strips each of width, to estimate the value of + d. 4] 5. Jan 008 qu.5 dy The gradient of a curve is given by = d. The curve passes through the point (4, 50). Find the equation of the curve. [6]

4 6. Jan 008 qu.7 The diagram shows part of the curve y = and the line = 5. 5 ( ) Eplain why d does not give the total area of the regions shaded in the 0 diagram. [] Use integration to find the eact total area of the shaded regions. [7] 7. June 007 qu.4 The diagram shows the curve y = 4 +. Use the trapezium rule, with strips of width 0.5, to find an approimate value for the area of the region bounded by the curve y = 4 +, the -ais, and the lines = and =.Give your answer correct to significant figures. [4] State with a reason whether this approimation is an under-estimate or an over-estimate. [] 8. June 007 qu.6 (a) Find ( 4) d. [] 6 Hence evaluate ( 4) d. [] 6 (b) Find d. []

5 9. Jan 007 qu. ( ) Find 4 5 d [] dy The gradient of a curve is given by = 4 5. The curve passes through the point d (, 7). Find the equation of the curve. [] 0. Jan 007 qu.5 (b) Use the trapezium rule, with two strips of width, to find an approimate value for 9 log 0 d giving your answer correct to significant figures. [4]. Jan 007 qu.0 The diagram shows the graph of y =. Verify that the curve intersects the -ais at (9, 0). [] The shaded region is enclosed by the curve, the -ais and the line = a (where a > 9).Given that the area of the shaded region is 4 square units, find the value of a. [9]. June 006 qu. The gradient of a curve is given by d y =, and the curve passes through the point (4, 5). d Find the equation of the curve. [6]

6 . June 006 qu.4 The diagram shows the curve y = 4 and the line y = +. Find the -coordinates of the points of intersection of the curve and the line. [] Use integration to find the area of the shaded region bounded by the line and the curve. [6] 4. June 006 qu.9 Sketch the curve y =, and state the coordinates of any point where the curve crosses an ais. [] Use the trapezium rule, with 4 strips of width 0.5, to estimate the area of the region bounded by the curve y =, the aes, and the line =. [4] (iii) The point P on the curve y = has y-coordinate equal to. 6 log0 Prove that the -coordinate of P may be written as +. [4] log 5. Jan 006 qu.6 0 (a) Find ( + 4) d. [4] a (b) Find the value, in terms of a, of 4 d, where a is a constant greater than. [] d Deduce the value of 4. []

7 6. Jan 006 qu.8 The cubic polynomial + k + 6 is denoted by f(). It is given that ( + ) is a factor of f(). Show that k = 5, and factorise f() completely. [6] Find f ( ) d. [4] (iii) Eplain with the aid of a sketch why the answer to part does not give the area of the region between the curve y = f() and the -ais for. [] 7. June 005 qu. Find ( + )( + ) d [4] 9 Evaluate d 0 []