## Trapezoidal Rule

Here, the integral is computed on each of the sub-intervals by using linear interpolating formula, i.e. for and then summing them up to obtain the desired integral.

Note that

Now using the formula ( 13.3.2) for on the interval we get,

Thus, we have,

i.e.
 (13.3.4)

This is called TRAPEZOIDAL RULE. It is a simple quadrature formula, but is not very accurate.

Remark 13.3.1   An estimate for the error in numerical integration using the Trapezoidal rule is given by

where is the average value of the second forward differences.

Recall that in the case of linear function, the second forward differences is zero, hence, the Trapezoidal rule gives exact value of the integral if the integrand is a linear function.

EXAMPLE 13.3.2   Using Trapezoidal rule compute the integral where the table for the values of is given below:
 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.01005 1.04081 1.09417 1.17351 1.28402 1.43332 1.63231 1.89648 2.2479 2.71828

Solution: Here,

and

Thus,

A K Lal 2007-09-12