is the half$-$width of the $-$subinterval

is the number of node points

are the weights

are the node points for the $-$subinterval.

Summing over all subintervals yields the numerical approximation of the integral $.$ A popular quadrature method of this form is Simpson's $1/3$ rule. Here, three uniformly spaced node points are used; the endpoints and the midpoint of the subinterval. The quadrature rule is:

where is the midpoint of the subinterval. Note that for this rule, the halfwidth of the subinterval is also the spacing between the node points within the subinterval.

Write a function that implements Simpson's $1/3$ rule using uniformly spaced subintervals without using a loop. Your function should accept $4$ inputs:

fun $=$ a handle to the integrand function

a $=$ lower limit of integration

b $=$ upper limit of integration

n $=$ number of uniform subintervals

and return $1$ output:

Q $=$ value of the integral