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 overes