C3 Integration 1. June 2010 qu. 4

Transcription

1 C Integration. June qu. 4 k The diagram shows part of the curve y =, where k is a positive constant. The points A and B on the curve have -coordinates and 6 respectively. Lines through A and B parallel to the aes as shown meet at the point C. The region R is bounded by the curve and the lines =, = 6 and y =. The region S is bounded by the curve and the lines AC and BC. It is given that the area of the region R is ln 8. Show that k = 4. [] Find the eact volume of the solid produced when the region S is rotated completely about the -ais. [4]. June qu.7 The diagram shows the curve with equation y = ( ) 4. The point P on the curve has coordinates (, 6) and the tangent to the curve at P meets the -ais at the point Q. The shaded region is bounded by PQ, the -ais and that part of the curve for which. Find the eact area of this shaded region. []. Jan qu. Find ( 7) d. []

2 4. Jan qu. Find, in simplified form, the eact value of 6 d. [] Use Simpson s rule with two strips to find an approimation to 6 d. [] (iii) 5 Use your answers to parts and to show that ln. 6 [] 5. Jan qu.6 Given that ( ke + ( k )e ) d = 85, find the value of the constant k. [7] 6. June 9 qu. ln 4 The diagram shows the curve with equation y = ( ). The shaded region is bounded by the curve and the lines = and y =. Find the eact volume obtained when the shaded region is rotated completely about the -ais. [5] 7. June 9 qu. 4 It is given that a a (e + e )d =, where a is a positive constant. Show that a = ln( + e a e a ). 9 [5] Use an iterative process, based on the equation in part, to find the value of a correct to 4 decimal places. Use a starting value of.6 and show the result of each step of the process. 8. Jan 9 qu. Find 8e d, (4 + 5) 6 d. [5] 9. Jan 9 qu.8 The diagram shows the curve with equation y = 6. The point P has coordinates (, p). The shaded region is bounded by the curve and the lines =, y = and y = p. The shaded region is rotated completely about the y-ais to form a solid of volume V. 7 Show that V = 6π. [6] ( p + ) It is given that P is moving along the y-ais in such a way that, at time t, the variables p and t d p dv are related by = p +. Find the value of at the instant when p = 9. [4] dt dt

3 . June 8 qu. 6 The diagram shows the curves y = e and y = ( ) 4. The shaded region is bounded by the two curves and the line =. The shaded region is rotated completely about the -ais. Find the eact volume of the solid produced. [9]. Jan 8 qu.5 9 Find ( + 7) d. [] (b). June 7 qu. 4 The diagram shows the curve y =. The shaded region is bounded by the curve and the lines =, = 6 and y =. The shaded region is rotated completely about the -ais. Find the eact volume of the solid produced, simplifying your answer. [5] ( The integral I is defined by I = + ) d. Use integration to find the eact value of I. [4] Use Simpson s rule with two strips to find an approimate value for I. Give your answer correct to significant figures. []. June 7 qu. 6 a Given that ( 6e + )d = 4, show that a = ln(5 a ). 6 [5] Use an iterative formula, based on the equation in part, to find the value of a correct to decimal places. Use a starting value of and show the result of each iteration. [4]

4 4. June 7 qu. 8 4ln dy 4 Given that y =, show that =. 4ln + d (4ln + ) [] 4ln Find the eact value of the gradient of the curve y = 4ln + at the point where it crosses the -ais. [4] (iii) y 5. Jan 7 qu.6 O e The diagram shows part of the curve with equation y =. (4ln + ) The region shaded in the diagram is bounded by the curve and the lines =, = e and y =. Find the eact volume of the solid produced when this shaded region is rotated completely about the -ais. [4] The diagram shows the curve with equation y =. The shaded region is bounded by the + curve and the lines =, = and y =. Find the eact area of the shaded region. [4] The shaded region is rotated completely about the -ais. Find the eact volume of the solid formed, simplifying your answer. [5] 6. June 6 qu. 7 Find the eact value of (4 ) d. [4] (b) The diagram shows part of the curve y =. The point P has coordinates a, and the point Q has a coordinates a,, where a is a positive constant. The point R is such that PR is parallel to the -ais a and QR is parallel to the y-ais. The region shaded in the diagram is bounded by the curve and by the lines PR and QR. Show that the area of this shaded region is ln( e ). [6]

5 7. June 6 qu. 9 y y = ln( ) P O The diagram shows the curve with equation y = ln( ). The point P has coordinates (, p). The region R, shaded in the diagram, is bounded by the curve and the lines =, y = and y = p. The units on the aes are centimetres. The region R is rotated completely about the y-ais to form a solid. Show that the volume, V cm, of the solid is given byv = π(e P + 4e + p 5). [8] It is given that the point P is moving in the positive direction along the y-ais at a constant rate of. cm min. Find the rate at which the volume of the solid is increasing at the instant when p = 4, giving your answer correct to significant figures. [5] 8. Jan 6 qu. Show that 8 d = ln 64. ` [4] 9. Jan 6 qu.5 P The diagram shows the curves y = ( ) 5 and y = e. The curves meet at the point (, ). Find the eact area of the region (shaded in the diagram) bounded by the y-ais and by part of each curve. [8]. June 5 qu. 4 The diagram shows the curve y =. The region R, shaded in the diagram, is bounded by the curve and by the lines =, = 5 and y =. The region R is rotated completely about the -ais. Find the eact volume of the solid formed. [4] 5 (b) Use Simpson s rule, with 4 strips, to find an approimate value for ( + ) d, giving your answer correct to decimal places. [4]