# 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.

A solution for the final is available.

Department of Mathematics, PO Box 643113, Neill Hall 103, Washington State University, Pullman WA 99164-3113, 509-335-3926, Contact Us