# The Midpoint Rule

One alternative to the composite trapezoidal rule for integrating functions is to use the midpoint rule. The midpoint rule is simply a Riemann sum in which the points where the function is evaluated are the midpoints of subintervals of the interval of integration. Specifically, consider the integral \[\int_a^b f(x)\,dx\] where \(f\) is twice continuously differentiable. Let \(a=x_0\lt x_1\lt\dots\lt x_n=b\) be an equally-spaced partition of \([a,b]\), and let \(z_i=(x_i+x_{i-1})/2\) for \(i=1,2,\dots,n\) denote the midpoints of the subintervals defined by the \(x_i\) values. When \(h=(b-a)/n\) is the uniform width of the subintervals, the midpoint rule is \[\int_a^b f(x)\,dx = h\sum_{i=1}^n f(z_i) + O(h^2). \] Write a Matlab function that takes four arguments: an integrand function, the lower and upper limits of integration, and the value of \(n\), which returns the value of the midpoint approximation to the integral.

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