DHS Trapezium and Simpson's Rule (9649) Topical Revision

Numerical Methods – Trapezium and Simpson’s Rule NUMERICAL METHODS [FM] TRAPEZIUM AND SIMPSON’S RULE 1 The function f is given by for . (i) Use the trapezium rule with 4 equal intervals to estimate . [3] (ii) Explain why the trapezium rule un derestimates the value of . [1] (iii) Use the Simpson’s rule with 4 equal intervals to estimate . [3] (iv) Explain why the Simpson’s rule gives a be tter estimate than the trapezium rule. [1] (v) Use the trapezium rule with the same interval width as (i) to estimate . [2] [EJC/FM/2019/P2/Q3] 2 Let f be a function such that    ff x x  for 0 xa where a is a positive constant. The integral I is defined by  2 0 f d a Ix a b x  where b is a constant. (i) Use Trapezium rule with 3 ordinates, estimate I in terms of a and b. [3] (ii) Use your answer in (i) to estimate the value of 6 0 3t a n d 12xx    . [1] (iii) Show that the estimate in (ii) is the exact value of 6 0 3t a n d 12xx    . [2] [TJC/FM/2019/MYE/Q3] 3 (i) Using the Simpson’s rule with 5 ordinates, find an approximation to 2 3 43 0 23 d ,xxx x leaving your answer to 3 decimal places. [3] (ii) Hence, without the use of a calculator, find an approximation to 2 33 4 2 23 d ,x xx x   leaving your answer to 3 decimal places. Justify your answer. [2] [NJC/FM/2018/MYE/Q2] 2f( ) (4 ) ( 8 )x xx x x 4 0 f( )dx x 4 0 f( )dx x 4 0 f( )dx x 8 0 f( )dx x H2 Double Math

Numerical Methods – Trapezium and Simpson’s Rule 4 (a) The arc of the curve  ln cosyx , from the point where 0x to the point where 6x  , is denoted by C. Show that S, the area of the surface generated when C is rotated through one revolution about the x-axis, is 6 0 2s e c l n s e c d .x xx   [3] Use the trapezium rule with 5 ordinates to find an approximation for the integral 6 0 sec ln sec dx xx   , giving your answer correct to 4 decimal places. Explain, with the aid of a sketch, whether the es timate is an over-estimate or an under- estimate. [5] (b) The region R is bounded by the curves 42 42 yyx and 2 2 yx , from the point where 0y to the point where 2y . Sketch the two curves in a single diagram, indicating clearly the region R. Find the exact volume of the solid formed when R is rotated completely about the line 5 8y , giving your answer in the form k , where k is a constant to be determined. [6] [RI/FM/2019/MCT/Q6] 5 (i) By using the trapezium rule, with 6 or dinates, find an approximate value for 2 0 sin d2 xx x    , giving your answer correct to 3 decimal places. Explain, with the aid of a sketch, why the value obtained is an overestimate. [3] (ii) Explain the feasibility in applying Simps on’s rule to the ordinates used in part (i). [1] (iii) Use Simpson’s rule, with 4 intervals of equal wi dth, to find an estimate for the integral in part (i), leaving your answer correct to 3 decimal places. [2] (iv) Explain why Simpson's rule usually give s a more accurate answer than the trapezium rule. [1] [HCI et al/FM/2018/P1/Q7b] 6 By considering an appropriate area under the curve 2 1y x for 0x , use the trapezium rule with strips of unit width to show that 1 22 2 11 1 1 22 n rnn r      . [3] Prove that the area of trapezium bounded by the x-axis, the lines 1 2x r , and the tangent to the curve 2 1y x at the point 2 1,r r   is 2 1 r . [3]

Numerical Methods – Trapezium and Simpson’s Rule By considering an appropriate area under the curve 2 1y x for 0x , show that 1 2 2 122 32 1 n r rn     . [3] Hence, without using a gra phic calculator, show that 22 2 11 11.49004 1 ... 1.6567223 1 0 0     . [2] [ACJC/FM/2018/P2/Q5] 7 Let 1 20 1 d1Ix x  . An estimate of I denoted by nI is obtained by using the trapezium rule with n trapeziums each of equal width. (i) Find 3I . [2] (ii) It is given that 2 1 nII k n   , where k is a constant independent of n. By taking 4 0.7828I  and using your value of 3I , find a better approximation to I, leaving your answer to four decimal places. [3] Hence, find an approximation for  , giving your answer to four decimal places. [2] [TJC/FM/2019/P1/Q2] 8 The mid-point rule using one strip to approximate the integral f( ) d b a x x is given by f( ) d ( )f . 2 b a abxx b a   The integral 2 0 f( ) d , for a given f ( ),Ix x x is being evaluated numerically by the mid-point rule and the trapezium rule. The following estimates to 6 decimal places have been obtained. Number of strips Mid-point rule Trapezium rule 1 3.464102 3.650282 2 3.510411 IT (i) Show that IT = 3.557192. [3] (ii) Use Simpson’s rule with two strips to determine an estimate of I. [2] (iii) Give a value of I to a degree of accuracy that appears justified. [1] [VJC/FM/2019/P1/Q2]

Numerical Methods – Trapezium and Simpson’s Rule 9 (i) Find the exact value of 1 20 1 d1 xx . [2] (ii) Use the trapezium rule with 7 ordinates to find an estimate for 1 20 1 d1 xx , leaving your answer correct to 8 decimal places. [3] (iii) Use Simpson’s rule, with 6 interva ls of equal width, to find an estimate for 1 20 1 d1 xx , leaving your answer correct to 8 decimal places. [2] The Simpson’s 3 8 rule is another numerical integratio n method that is based on a cubic interpolation rather than a quadratic interpolat ion. This method can be used when the number of intervals are multiples of 3. The Simpson’s 3 8 rule states that where bah n  , 133 3 03 23 13 111 3f ( ) d f ( ) 3f ( ) 3f ( ) 2f () f ( )8 nn n b iii na iii hx xx x x x x          (iv) Use Simpson’s 3 8 rule, with 6 intervals of equal width, to find an estimate for 1 20 1 d1 xx , leaving your answer correct to 8 decimal places. [3] (v) Compare how well the above th ree methods provide estimates for 1 20 1 d1 xx . Give an advantage of the Simpson’s 3 8 rule as compared to the Simpson’s rule. [3] [HCI/FM/2019/MCT/Q7] 10 (a) Using Simpson’s rule with 3 ordinates, obtain a value for 0 0 4 d xh xh x x   , where h is a positive constant, and show that the value you obtain is in error by 5kh , where k is a positive constant to be determined exactly. [5] (b) The differential equation 2d 2d y xy yx  is to be solved numerically using the second order Taylor series      2 00 00 2! hyx h yx h y x y x     . Given that  01y  and taking 0.2h , calculate successively the value of  0.2y and  0.4y . [4] [NJC/FM/2019/P1/Q6]

Numerical Methods – Trapezium and Simpson’s Rule 11 Explain why Simpson’s rule usually gives a better approximation than trapezium rule does. [2] A curve has equation f( ) s i n , 0 πyx xx   . (i) Write down a definite integral for the length of the curve. [1] (ii) By using a calculator, evaluate, correct to 10 significant figures,