In view of this, we have
To find the error estimates, we consider the forward difference table, which is given below:
Thus, error due to Trapezoidal rule is,
Solution: Note that here the points are not given to be
equidistant, so as such we can not use any of the above two
formulae. However, we notice that the tabular points
are equidistant and so are the tabular points
. Now we can
divide the interval in two subinterval:
. The integrals then can be evaluated in each interval. We observe that the second set has odd number of points. Thus, the first integral is evaluated by using Trapezoidal rule and the second one by Simpson's rule (of course, one could have used Trapezoidal rule in both the subintervals).
It may be mentioned here that in the above integral, and that the value of the integral is . It will be interesting for the reader to compute the two integrals using Trapezoidal rule and compare the values.