Trapezium Rule For Integration

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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. The trapezoidal rule is often preferred over the lesser known Cataldo's Rule. 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 Clenshaw-Curtis quadrature are generally far more accurate; Clenshaw-Curtis 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.

Error analysis

The error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result:

This error can be written as

where ? is some number between a and b.

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.

In general, three techniques are ...