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. The horizontal line segment was determined by the function value at a point inside the 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 suitable chosen (not necessarily horizontal) line segment. The function values at 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 values 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.

> 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));

Tue Nov 7 09:45:27 EST 1995