17 - Definite integrals Trapezoidal rule Simpsons rules...

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Unformatted text preview: Definite integrals Trapezoidal rule Simpsons rules Numerical Integration Formulas Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Numerical Integration Formulas MATH 2070U 1 / 32 Definite integrals Trapezoidal rule Simpsons rules Numerical Integration Formulas 1 Definite integrals 2 Trapezoidal rule Composite trapezoidal rule 3 Simpsons rules Composite Simpsons 1/3 rule Simpsons 3 8 rule c D. Aruliah (UOIT) Numerical Integration Formulas MATH 2070U 2 / 32 Definite integrals Trapezoidal rule Simpsons rules Definite integrals Recall definite integral I ( f ) : = I ( f ; a , b ) = Z b a f ( x ) dx is (signed) area under graph of y = f ( x ) between x = a and x = b Quadrature or numerical integration : numerical approximation of definite integrals c D. Aruliah (UOIT) Numerical Integration Formulas MATH 2070U 4 / 32 Definite integrals Trapezoidal rule Simpsons rules Evaluation/computation of definite integrals The Fundamental theorem of calculus: I ( f ) = Z b a f ( x ) dx = F ( b )- F ( a ) where F is an antiderivative or primitive of f Definite integral R b a f ( x ) dx is a number Indefinite integral R f ( x ) dx is a function Many integrands do not admit analytic antiderivatives e.g. erf ( x ) : = 2 Z x e- t 2 dt (the error function) e.g. E ( k ) : = Z 2 p 1- k 2 sin 2 d (complete elliptic integral) c D. Aruliah (UOIT) Numerical Integration Formulas MATH 2070U 5 / 32 Definite integrals Trapezoidal rule Simpsons rules Quadrature rules Quadrature : numerical approximation of definite integrals Quadrature rules have basic form I appr ( f ) n k = w k f ( x k ) = w f ( x ) + w 1 f ( x 1 ) + + w n- 1 f ( x n- 1 ) + w n f ( x n ) Replaces integration by weighted sum of function evaluations { x k } n k = are quadrature nodes/points { w k } n k = are quadrature weights Weights depend on { x k } n k = , width b- a of interval Rules derived by integrating polynomial interpolants Newton-Cotes rules use equally spaced quadrature points c D. Aruliah (UOIT) Numerical Integration Formulas MATH 2070U 6 / 32 Definite integrals Trapezoidal rule Simpsons rules Trapezoidal rule Trapezoidal rule The trapezoidal rule for approximating R b a f ( x ) dx is given by I : = b- a 2 [ f ( a ) + f ( b )] . Sample f at end points x = a , b of interval [ a , b ] Approximate R b a f ( x ) dx by trapezoid Area = ( width )( average height ) = ( b- a ) f ( a ) + f ( b ) 2 c D. Aruliah (UOIT) Numerical Integration Formulas MATH 2070U 8 / 32 Definite integrals Trapezoidal rule Simpsons rules c D. Aruliah (UOIT) Numerical Integration Formulas MATH 2070U 9 / 32 Definite integrals Trapezoidal rule Simpsons rules Trapezoidal rule: example Approximate Z 1 p 1 + x 4 dx using the trapezoidal rule....
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17 - Definite integrals Trapezoidal rule Simpsons rules...

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