Two widely used rules for approximating areas are the trapezoidal rule and Simpson's rule. To motivate the new methods, we recall that rectangular rules approximated the function by a horizontal line in each interval. It is reasonable to expect that if we approximate the function more accurately inside each interval then a more efficient numerical scheme will follow. This is the idea behind the trapezoidal and Simpson's rules. Here the trapezoidal rule approximates the function by a suitably chosen (not necessarily horizontal) line segment. The function values at the two points in the interval are used in the approximation. While Simpson's rule uses a suitably chosen parabolic shape (see Section 4.6 of the text) and uses the function at three points.

The Maple `student` package has commands `
trapezoid` and `simpson` that implement these methods. The command
syntax is very similar to the rectangular approximations. See the
examples below. Note that an even number of subintervals is required
for the `simpson` command and that the default number of
subintervals is *n*=4 for both `trapezoid` and `simpson`.

> with(student):

> trapezoid(x^2,x=0..4);

> evalf(trapezoid(x^2,x=0..4));

> evalf(trapezoid(x^2,x=0..4,10));

> simpson(x^2,x=0..4);

> evalf(simpson(x^2,x=0..4));

> evalf(simpson(x^2,x=0..4,10));

Mon Nov 11 16:16:00 EST 1996