Regent College Maths Department. Core Mathematics 4 Trapezium Rule. C4 Integration Page 1

Size: px

Start display at page:

Download "Regent College Maths Department. Core Mathematics 4 Trapezium Rule. C4 Integration Page 1"

Moses Bridges
5 years ago
Views:

1 Regent College Maths Department Core Mathematics Trapezium Rule C Integration Page

2 Integration It might appear to be a bit obvious but you must remember all of your C work on differentiation if you are to succeed with this unit. Content Use of trapezium rule to provide approimate solution to integrals. C Integration Page

3 Standard Integrals Function (f()) n (a + b) n a b e Sin Cos Tan Cot Cosec Sec Sec Sec Tan -Cosec e a+b Integral f()d n c n ln c n (a b) c a(n ) ln a b c a e + c -Cos + c Sin + c ln (Sec ) + c ln (Sin ) + c -ln (Cosec + Cot ) + c ln (Sec + Tan ) + c Tan + c Sec + c Cot + c e a ab c Cos (a + b) Sin(a b) c a Sin n Cos n Sin c n Cos n Sin n Cos c n Sin Sin c Cos Sin c C Integration Page

4 C Integration Page Eample The diagram below is the graph of the function y = Sec Use the trapezium rule with five equally spaced ordinates to estimate the area of the region bounded by the curve with equation y = sec, the -ais and the lines = - and =, giving your answer to two decimal places. The five coordinates are and 6 0 6,,,. The best way to succeed with these questions is to put the information into a table Y Trapezium rule is introduced in AS. b a n n o h d f )... ( ) ( ) (... dp Secd

5 Past paper questions on Trapezium rule. Figure y R Figure shows the graph of the curve with equation y = e, 0. The finite region R bounded by the lines =, the -ais and the curve is shown shaded in Figure. (a) Use integration to find the eact value of the area for R. (b) Complete the table with the values of y corresponding to = 0. and 0.8. (5) y = e (c) Use the trapezium rule with all the values in the table to find an approimate value for this area, giving your answer to significant figures. () () (C, June 005 Q5) C Integration Page 5

6 . (a) Given that y = sec, complete the table with the values of y corresponding to =, and y.069 () (b) Use the trapezium rule, with all the values for y in the completed table, to obtain an estimate for sec d. Show all the steps of your working and give your answer to decimal places. 0 () The eact value of sec d is ln ( + ). 0 (c) Calculate the % error in using the estimate you obtained in part (b). () (C, Jan 006 Q). Figure y O Figure shows a sketch of the curve with equation y = ( ) ln, > 0. (a) Copy and complete the table with the values of y corresponding to =.5 and =.5. C Integration Page 6

7 y e e e ().5.5 Given that I = y 0 ln ln ( ) ln d, (b) use the trapezium rule () (i) with values at y at =, and to find an approimate value for I to significant figures, (ii) with values at y at =,.5,,.5 and to find another approimate value for I to significant figures. (5) (c) Eplain, with reference to Figure, why an increase in the number of values improves the accuracy of the approimation. () (d) Show, by integration, that the eact value of (. I = e 5 0 ) ( ) ln d is ln. d. (6) (C, June 006 Q6) (a) Given that y = e ( + ), copy and complete the table with the values of y corresponding to =, and (b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the original integral I, giving your answer to significant figures. () b t (c) Use the substitution t = ( + ) to show that I may be epressed as kte dt, giving the a values of a, b and k. (5) (d) Use integration by parts to evaluate this integral, and hence find the value of I correct to significant figures, showing all the steps in your working. (5) (C, Jan 007 Q8) C Integration Page 7

8 Figure Figure shows part of the curve with equation y = (tan ). The finite region R, which is bounded by the curve, the -ais and the line =, is shown shaded in Figure. (a) Given that y = (tan ), copy and complete the table with the values of y corresponding to =, and, giving your answers to 5 decimal places y 0 (b) Use the trapezium rule with all the values of y in the completed table to obtain an estimate for the area of the shaded region R, giving your answer to decimal places. () The region R is rotated through radians around the -ais to generate a solid of revolution. (c) Use integration to find an eact value for the volume of the solid generated. () () (C, June 007 Q7) 6. C Integration Page 8

9 Figure The curve shown in Figure has equation e (sin ), 0. The finite region R bounded by the curve and the -ais is shown shaded in Figure. (a) Copy and complete the table below with the values of y corresponding to = and =, giving your answers to 5 decimal places. 0 y (b) Use the trapezium rule, with all the values in the completed table, to obtain an estimate for the area of the region R. Give your answer to decimal places. () () (C, Jan 008 Q) 7. C Integration Page 9

10 Figure 0.5 Figure shows part of the curve with equation y = e. The finite region R, shown shaded in Figure, is bounded by the curve, the -ais, the y-ais and the line =. (a) Copy and complete the table with the values of y corresponding to = 0.8 and = y e 0 e 0.08 e 0.7 e (b) Use the trapezium rule with all the values in the table to find an approimate value for the area of R, giving your answer to significant figures. () (C, June 008 Q) () 8. C Integration Page 0

Figure Figure shows the finite region R bounded by the -ais, the y-ais and the curve with equation y = cos, 0. The table shows corresponding values of and y for y = cos. 0 8 9 8 y.776.

11 Figure Figure shows the finite region R bounded by the -ais, the y-ais and the curve with equation y = cos, 0. The table shows corresponding values of and y for y = cos y (a) Copy and complete the table above giving the missing value of y to 5 decimal places. (b) Using the trapezium rule, with all the values of y from the completed table, find an approimation for the area of R, giving your answer to decimal places. () (c) Use integration to find the eact area of R. () () (C, June 009 Q) 9. C Integration Page

12 Figure Figure shows a sketch of the curve with equation y = ln,. The finite region R, shown shaded in Figure, is bounded by the curve, the -ais and the line =. The table shows corresponding values of and y for y = ln y (a) Copy and complete the table with the values of y corresponding to = and =.5, giving your answers to decimal places. () (b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to decimal places. () (c) (i) Use integration by parts to find ln d. (ii) Hence find the eact area of R, giving your answer in the form (a ln + b), where a and b are integers. (7) (C, Jan 00 Q) 0. C Integration Page

Figure Figure shows part of the curve with equation y = (0.75 + cos ). The finite region R, shown shaded in Figure, is bounded by the curve, the y-ais, the -ais and the line with equation =.

13 Figure Figure shows part of the curve with equation y = ( cos ). The finite region R, shown shaded in Figure, is bounded by the curve, the y-ais, the -ais and the line with equation =. (a) Copy and complete the table with values of y corresponding to = 6 and =. 0 6 y.9.97 (b) Use the trapezium rule () (i) with the values of y at = 0, = and = to find an estimate of the area of R. 6 Give your answer to decimal places. (ii) with the values of y at = 0, =, =, = and = to find a further estimate 6. of the area of R. Give your answer to decimal places. (6) (C, June 00 Q) C Integration Page

14 Figure Figure shows a sketch of the curve with equation y = ln ( + ), 0. The finite region R, shown shaded in Figure, is bounded by the curve, the -ais and the line =. The table below shows corresponding values of and y for y = ln ( + ). 0 y (a) Complete the table above giving the missing values of y to decimal places. (b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to decimal places. () (c) Use the substitution u = + to show that the area of R is (). (d) Hence, or otherwise, find the eact area of R. ( u )ln u du. () (6) (C, June 0 Q) C Integration Page

15 sin Figure shows a sketch of the curve with equation y =, 0. ( cos ) The finite region R, shown shaded in Figure, is bounded by the curve and the -ais. sin The table below shows corresponding values of and y for y =. ( cos ) y (a) Complete the table above giving the missing value of y to 5 decimal places. (b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to decimal places. () (c) Using the substitution u = + cos, or otherwise, show that (). where k is a constant. sin d ( cos ) = ln ( + cos ) cos + k, (d) Hence calculate the error of the estimate in part (b), giving your answer to significant figures. (C, Jan0 Q6) (5) C Integration Page 5

Figure Figure shows a sketch of part of the curve with equation y = ln. The finite region R, shown shaded in Figure, is bounded by the curve, the -ais and the lines = and =.

16 Figure Figure shows a sketch of part of the curve with equation y = ln. The finite region R, shown shaded in Figure, is bounded by the curve, the -ais and the lines = and =. (a) Use the trapezium rule, with strips of equal width, to find an estimate for the area of R, giving your answer to decimal places. () (b) Find ln d. (c) Hence find the eact area of R, giving your answer in the form a ln + b, where a and b are eact constants. () (C, June0 Q7) (). C Integration Page 6

17 Figure Figure shows a sketch of part of the curve with equation y =. The finite region R, shown shaded in Figure, is bounded by the curve, the -ais, the line with equation = and the line with equation =. (a) Copy and complete the table with the value of y corresponding to =, giving your answer to decimal places. () y (b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate of the area of the region R, giving your answer to decimal places. () (c) Use the substitution u = +, to find, by integrating, the eact area of R. (8) (C, Jan0 Q) 5. C Integration Page 7

Figure Figure shows the finite region R bounded by the -ais, the y-ais, the line = and the curve with equation y = sec, 0 The table shows corresponding values of and y for y = sec. 0 6 y.0576.

18 Figure Figure shows the finite region R bounded by the -ais, the y-ais, the line = and the curve with equation y = sec, 0 The table shows corresponding values of and y for y = sec. 0 6 y (a) Complete the table above giving the missing value of y to 6 decimal places. (b) Using the trapezium rule, with all of the values of y from the completed table, find an approimation for the area of R, giving your answer to decimal places. () Region R is rotated through radians about the -ais. () 6. (c) Use calculus to find the eact volume of the solid formed. () (C, June0Q) C Integration Page 8

Figure Figure shows part of the curve with equation e. The finite region R shown shaded in Figure is bounded by the curve, the -ais, the t-ais and the line t = 8.

19 Figure Figure shows part of the curve with equation e. The finite region R shown shaded in Figure is bounded by the curve, the -ais, the t-ais and the line t = 8. t t (a) Complete the table with the value of corresponding to t = 6, giving your answer to decimal places. t (b) Use the trapezium rule with all the values of in the completed table to obtain an estimate for the area of the region R, giving your answer to decimal places. () (c) Use calculus to find the eact value for the area of R. (d) Find the difference between the values obtained in part (b) and part (c), giving your answer to decimal places. () () (6) (C, June0_R, Q5) 7. C Integration Page 9

20 Figure Figure shows a sketch of part of the curve with equation 0 y 5, > 0. The finite region R, shown shaded in Figure, is bounded by the curve, the -ais, and the lines with equations = and =. The table below shows corresponding values of and y for 0 y 5. y (a) Complete the table above by giving the missing value of y to 5 decimal places. (b) Use the trapezium rule, with all the values of y in the completed table, to find an estimate for the area of R, giving your answer to decimal places. () (c) By reference to the curve in Figure, state, giving a reason, whether your estimate in part (b) is an overestimate or an underestimate for the area of R. () (d) Use the substitution u =, or otherwise, to find the eact value of () 0 d 5 (6) (C, June0, Q) C Integration Page 0

8. Figure Figure shows a sketch of part of the curve with equation y = ( )e, The finite region R, shown shaded in Figure, is bounded by the curve, the -ais and the y-ais.

21 8. Figure Figure shows a sketch of part of the curve with equation y = ( )e, The finite region R, shown shaded in Figure, is bounded by the curve, the -ais and the y-ais. The table below shows corresponding values of and y for y = ( )e y (a) Use the trapezium rule with all the values of y in the table, to obtain an approimation for the area of R, giving your answer to decimal places. () (b) Eplain how the trapezium rule can be used to give a more accurate approimation for the area of R. () (c) Use calculus, showing each step in your working, to obtain an eact value for the area of R. Give your answer in its simplest form. (5) (C, June0_R, Q) C Integration Page

(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)

. f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula