  Simple proofs of the midpoint, trapezoidal and Simpson's rules are proved for
numerical integration on a compact interval. The integrand is assumed to be
twice continuously differentiable for the midpoint and trapezoidal rules, and
to be four times continuously differentiable for Simpson's rule. Errors are
estimated in terms of the uniform norm of second or fourth derivatives of the
integrand. The proof uses only integration by parts, applied to the second or
fourth derivative of the integrand, multiplied by an appropriate polynomial or
piecewise polynomial function. A corrected trapezoidal rule that includes the
first derivative of the integrand at the endpoints of the integration interval
is also proved in this manner, the coefficient in the error estimate being
smaller than for the midpoint and trapezoidal rules. The proofs are suitable
for presentation in a calculus or elementary numerical analysis class. Several
student projects are suggested.
