Chapter 07.03 Simpson’s 1/3 Rule of Integration After reading this chapter, you should be able to 1. derive the formula for Simpson’s 1/3 rule of integration, 2. use Simpson’s 1/3 rule it to solve integrals, 3. develop the formula for multiple-segment Simpson’s 1/3 rule of integration, . use multiple-segment Simpson’s 1/3 rule of integration to solve integrals, and !. derive the true error formula for multiple-segment Simpson’s 1/3 rule. What is integration? Integration is the process of measuring the area under a function plotted on a graph. Why would we want to integrate a function? Among the most common examples are finding the velocity of a body from an acceleration function, and displacement of a body from a velocity function. Throughout many engineering fields, there are (what sometimes seems lie! countless applications for integral calculus. "ou can read about some of these applications in #hapters $%.$$A&$%.$$'. (ometimes, the evaluation of expressions involving these integrals can become daunting, if not indeterminate. )or this reason, a wide variety of numerical methods has been developed to simplify the integral. *ere, we will discuss (impson+s ,-. rule of integral approximation, which improves upon the accuracy of the trape/oidal rule. *ere, we will discuss the (impson+s ,-. rule of approximating integrals of the form ( ) ∫ · b a d" " f # where ! (" f is called the integrand, · a lower limit of integration · b upper limit of integration Simpson’s 1/3 Rule The trape/oidal rule was based on approximating the integrand by a first order polynomial, and then integrating the polynomial over interval of integration. (impson+s ,-. rule is an $%.$.., $%.$..0 #hapter $%.$. extension of Trape/oidal rule where the integrand is approximated by a second order polynomial. Figure 1 Integration of a function 1ethod ,2 *ence ∫ ∫ ≈ · b a b a d" " f d" " f # ! ( ! ( 0 where ! ( 0 " f is a second order polynomial given by 0 0 , $ 0 ! ( " a " a a " f + + · . #hoose !!, ( , ( a f a , 0 , 0 , _ ¸ ¸ , _ ¸ ¸ + + b a f b a and !! ( , ( b f b as the three points of the function to evaluate , $ a , a and 0 a . 0 0 , $ 0 ! ( ! ( a a a a a a f a f + + · · 0 0 , $ 0 0 0 0 0 , _ ¸ ¸ + + , _ ¸ ¸ + + · , _ ¸ ¸ + · , _ ¸ ¸ + b a a b a a a b a f b a f 0 0 , $ 0 ! ( ! ( b a b a a b f b f + + · · (olving the above three e3uations for unnowns, , $ a , a and 0 a give 0 0 0 0 $ 0 ! ( ! ( 0 4 ! ( ! ( b ab a a f b a abf b a abf b abf b f a a + − + + , _ ¸ ¸ + − + · 0 0 , 0 ! ( 0 4 ! ( . ! ( . 0 4 ! ( b ab a b bf b a bf a bf b af b a af a af a + − + , _ ¸ ¸ + − + + , _ ¸ ¸ + − − · 0 0 0 0 ! ( 0 0 ! ( 0 b ab a b f b a f a f a + − , _ ¸ ¸ + , _ ¸ ¸ + − · Then ∫ ≈ b a d" " f # ! ( 0 ( ) ∫ + + · b a d" " a " a a 0 0 , $ b a " a " a " a 1 ] 1 ¸ + + · . 0 . 0 0 , $ (impson+s ,-. 5ule of Integration $%.$... . 0 ! ( . . 0 0 0 , $ a b a a b a a b a − + − + − · (ubstituting values of , $ a , a and 0 a give 1 ] 1 ¸ + , _ ¸ ¸ + + − · ∫ ! ( 0 4 ! ( 6 ! ( 0 b f b a f a f a b d" " f b a (ince for (impson ,-. rule, the interval [ ] b a, is broen into 0 segments, the segment width 0 a b h − · *ence the (impson+s ,-. rule is given by 1 ] 1 ¸ + , _ ¸ ¸ + + ≈ ∫ ! ( 0 4 ! ( . ! ( b f b a f a f h d" " f b a (ince the above form has ,-. in its formula, it is called (impson+s ,-. rule. 1ethod 02 (impson+s ,-. rule can also be derived by approximating ! (" f by a second order polynomial using 7ewton+s divided difference polynomial as , _ ¸ ¸ + − − + − + · 0 ! ( ! ( ! ( 0 , $ 0 b a " a " b a " b b " f where ! ( $ a f b · a b a a f b a f b − + − , _ ¸ ¸ + · 0 ! ( 0 , a b a b a a f b a f b a b b a f b f b − − + − , _ ¸ ¸ + − + − , _ ¸ ¸ + − · 0 ! ( 0 0 0 ! ( 0 Integrating 7ewton+s divided difference polynomial gives us ∫ ∫ ≈ b a b a d" " f d" " f ! ( ! ( 0 ∫ 1 ] 1 ¸ , _ ¸ ¸ + − − + − + · b a d" b a " a " b a " b b 0 ! ( ! ( 0 , $ b a " b a a " b a " b a" " b " b 1 ] 1 ¸ , _ ¸ ¸ + + + − + , _ ¸ ¸ − + · 0 ! ( 4 ! . ( . 0 0 . 0 0 , $ $%.$..4 #hapter $%.$. , _ ¸ ¸ − + + − + − − + , _ ¸ ¸ − − − + − · 0 ! !( ( 4 ! !( . ( . ! ( 0 ! ( 0 0 . . 0 0 0 , $ a b b a a a b b a a b b a b a a b b a b b (ubstituting values of , $ b , , b and 0 b into this e3uation yields the same result as before 1 ] 1 ¸ + , _ ¸ ¸ + + − ≈ ∫ ! ( 0 4 ! ( 6 ! ( b f b a f a f a b d" " f b a 1 ] 1 ¸ + , _ ¸ ¸ + + · ! ( 0 4 ! ( . b f b a f a f h 1ethod .2 8ne could even use the 9agrange polynomial to derive (impson+s formula. 7otice any method of three&point 3uadratic interpolation can be used to accomplish this tas. In this case, the interpolating function becomes ! ( 0 ! ( 0 ! ( 0 0 0 ! !( ( ! ( ! ( 0 ! ( 0 ! ( 0 b f b a b a b b a " a " b a f b b a a b a b " a " a f b a b a a b " b a " " f , _ ¸ ¸ + − − , _ ¸ ¸ + − − + , _ ¸ ¸ + , _ ¸ ¸ − + , _ ¸ ¸ − + − − + − , _ ¸ ¸ + − − , _ ¸ ¸ + − · Integrating this function gets b a b a b f b a b a b " b a a " b a " b a f b b a a b a ab" " b a " a f b a b a a " b a b " b a " d" " f 1 1 1 1 1 1 1 1 1 1 ] 1 ¸ , _ ¸ ¸ + − − + + + − + , _ ¸ ¸ + , _ ¸ ¸ − + , _ ¸ ¸ − + + + − + − , _ ¸ ¸ + − + + + − · ∫ ! ( 0 ! ( 0 ! ( 4 ! . ( . 0 0 0 0 ! ( . ! ( ! ( 0 0 ! ( 4 ! . ( . ! ( 0 . 0 . 0 . 0 (impson+s ,-. 5ule of Integration $%.$..: ! ( 0 ! ( 0 ! !( ( 4 ! !( . ( . 0 0 0 ! ( 0 ! !( ( . ! ( ! ( 0 0 ! !( ( 4 ! !( . ( . 0 0 . . 0 0 . . 0 0 . . b f b a b a b a b b a a a b b a a b b a f b b a a b a a b ab a b b a a b a f b a b a a a b b a b a b b a a b , _ ¸ ¸ + − − − + + − + − − + , _ ¸ ¸ + , _ ¸ ¸ − + , _ ¸ ¸ − + − + − + − − + − , _ ¸ ¸ + − − + + − + − − · ;elieve it or not, simplifying and factoring this large expression yields you the same result as before 1 ] 1 ¸ + , _ ¸ ¸ + + − ≈ ∫ ! ( 0 4 ! ( 6 ! ( b f b a f a f a b d" " f b a 1 ] 1 ¸ + , _ ¸ ¸ + + · ! ( 0 4 ! ( . b f b a f a f h . 1ethod 42 (impson+s ,-. rule can also be derived by the method of coefficients. Assume ! ( 0 ! ( ! ( . 0 , b f c b a f c a f c d" " f b a + , _ ¸ ¸ + + ≈ ∫ 9et the right&hand side be an exact expression for the integrals , , ∫ b a d" , ∫ b a "d" and ∫ b a d" " 0 . This implies that the right hand side will be exact expressions for integrals of any linear combination of the three integrals for a general second order polynomial. 7ow . 0 , , c c c a b d" b a + + · − · ∫ b c b a c a c a b "d" b a . 0 , 0 0 0 0 + + + · − · ∫ 0 . 0 0 0 , . . 0 0 . b c b a c a c a b d" " b a + , _ ¸ ¸ + + · − · ∫ (olving the above three e3uations for , $ c , c and 0 c give 6 , a b c − · $%.$..6 #hapter $%.$. . ! ( 0 0 a b c − · 6 . a b c − · This gives ! ( 6 0 . ! ( 0 ! ( 6 ! ( b f a b b a f a b a f a b d" " f b a − + , _ ¸ ¸ + − + − ≈ ∫ 1 ] 1 ¸ + , _ ¸ ¸ + + − · ! ( 0 4 ! ( 6 b f b a f a f a b