In these simple approximation schemes, the area above each subinterval is approximated by the area of a rectangle, with the height of the rectangle being chosen according to some rule. The rules we will be concerned with are as follows.

**left endpoint rule**-
The height of the rectangle is the value of
the function
*f*(*x*) at the left-hand endpoint of the subinterval. **right endpoint rule**-
The height of the rectangle is the value of
the function
*f*(*x*) at the right-hand endpoint of the subinterval.

The Maple `student` package has commands for visualizing these
two rectangular area approximations. To use them, you first must
load the `student` package via the `with` command. Then try the
commands given below. Make sure you understand the differences between
the two different rectangular approximations.

> with(student):

> f:=x-> x^2 ;

> rightbox(f(x),x=0..4);

> leftbox(f(x),x=0..4);

There are also Maple commands `leftsum` and `rightsum` to sum the areas of the rectangles, see
the examples below. Note the use of `evalf` to obtain numerical answers.

> rightsum(f(x),x=0..4);

> evalf(rightsum(f(x),x=0..4));

> evalf(rightsum(x,x=0..2));

Unless specified, Maple will break the given interval into four subintervals. Below are some examples of how to change the number of subintervals used in the approximation.

> evalf(rightsum(x,x=0..2,10));

> evalf(rightsum(x,x=0..2,20));

> evalf(rightsum(x,x=0..2,100));

Tue Sep 10 14:03:56 EDT 1996