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
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.