https://www.math.vt.edu/people/qlfang/class_home/Lesson3.pdf

integral.f90

program integral
implicit none

! we will solve integral of x^2 using the Trapezoidal Rule
integer :: n
real :: a, b, step, delta, coeff, f0, fn, inner_total, x

! We integrate x^2 from 0 to 4 with n intervals
a = 0.0
b = 4.0
n = 8

! given n interval, the step is b/n
step = b / n
! print *, 'step ', step

! the formula looks like this
! delta(x)/2 * [f(x0) + 2f(x1) + 2f(x2) + ... + f(x_n)]
delta = (b-a) / n
coeff = delta / 2
!print *, 'delta ', delta
!print *, 'coeff ', coeff

! here we calulate f(x0) and f(x_n) where f(x) = x^2
f0 = a**2
fn = b**2

! now we calculate the rest of the intervals
! step in do-loop must be an integer, so iterate n-2 times
x = a + step
do n = 1, n - 2, 1
  x = x + step
  inner_total = inner_total + 2 * (x**2)
end do

print *, coeff * (f0 + inner_total + fn)
end program integral