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trapezoidal rule trapezoidal rule in numerical analysis the trapezoidal rule also known as the trapezoid rule or trapezium rule is an approximate technique for calculating the definite integral the trapezoidal rule is one of a family of formulas for numerical integration called newton­ cotes formulas of which the midpoint rule is similar to the trapezoid rule simpson s rule is another member of the same family and in general has faster convergence than the trapezoidal rule for functions which are twice continuously differentiable though not in all specific cases however for various classes of rougher functions ones with weaker smoothness conditions the trapezoidal rule has faster convergence in general than simpson s rule moreover the trapezoidal rule tends to become extremely accurate when periodic functions are integrated over their periods which can be analyzed in various ways for non-periodic functions however methods with unequally spaced points such as gaussian quadrature and clenshaw­curtis quadrature are generally far more accurate know more about partial fraction decomposition math.tutorvista.com page no 1/4

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clenshaw­curtis quadrature can be viewed as a change of variables to express arbitrary integrals in terms of periodic integrals at which point the trapezoidal rule can be applied accurately uniform grid for a domain discretized into n equally spaced panels or n+1 grid points 1 2 n+1 where the grid spacing is h b-a n the approximation to the integral becomes non-uniform grid when the grid spacing is non-uniform one can use the formula error analysis the error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result it follows that if the integrand is concave up and thus has a positive second derivative then the error is negative and the trapezoidal rule overestimates the true value this can also be seen from the geometric picture the trapezoids include all of the area under the curve and extend over it similarly a concave-down function yields an underestimate because area is unaccounted for under the curve but none is counted above if the interval of the integral being approximated includes an inflection point then the error is harder to identify even though the trapezoidal formula is considered to be less efficient in approximating definite integral it is found surprisingly efficient in some cases of periodic functions learn more rate of change math.tutorvista.com page no 2/4

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we may observe that the approximation using the trapezoidal rule for 20sinx dx will give the same value as the integral 0 but the trapezoidal approximation done for 0sinx dx will yield the difference consistent with the error bounds for the rule in the first cast the trapezoidal rule is applied to the function which is integrated over its full period in this case the graph will have one portion concave down in the interval 0 and another equal part of the graph is concave up in the interval 2 this phenomenon cancels the errors which occur when the approximation is done using trapezoidal rule but in the second case the integration is done over half period and hence the error occurred remain yielding not so accurate approximation when compared to simpson s rule the trapezoidal rule is less efficient in approximating a definite integral but the trapezoidal partitions done on the graph provides a clear visual explanation to the concept applied the formula is also easy to memorize and can be applied with ease math.tutorvista.com page no 4/4

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thank you for watching presentation

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