Hamed Nooraldin

In general, when a curve is concave down, the trapezoidal rule will underestimate the area because when you connect the trapezoid's left and right sides to the curve, and then connect those two points to make the trapezoid's top, you'll be left with a little space above the trapezoid.

The trapezoidal rule employs function values at nodes that are equally spaced. It is extremely precise for integrals over periodic intervals but is frequently wrong in nonperiodic circumstances.

The Trapezoidal Rule is the average of the left and right sums, and it typically provides a better estimate than either alone. Simpson's Rule approximates area by using intervals topped with parabolas; hence, it delivers the precise area beneath quadratic functions.

However, if the integrand is concave up (i.e. has a positive second derivative), the error is negative and the trapezoidal method overestimates the real value.

I suggest to see the following tutorials as well:

1. https://www.youtube.com/watch?v=ud3RQymhk0g

2. https://www.youtube.com/watch?v=MQx3Bo1uurM

3. https://www.youtube.com/watch?v=cz30Jaqtghk

thanks for your answer.

You mention you are learning fortran -- so let me offer a small hint

You have a loop from 1---100 yet you select with "if statements" certain values -- you appear to have no need to do the whole loop?

Why not make a list of key values of say K, stored in a pointer array KP = (2, 5, 11, ...., 100) and then call the subroutines with these index values by going over the elements in the array? You could get rid of most of the code.

I am learning and that is a good hint.

thank you very much.

see this link to learn trapezoidal rule

https://byjus.com/maths/trapezoidal-rule/