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. **midpoint rule**-
The height of the rectangle is the value of
the function
*f*(*x*) at the midpoint of the subinterval.

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

> with(student):

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

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

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

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

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

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

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

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

The Maple commands above use the short-hand summation notation. Since there are only four terms in the sums above, it isn't hard to write them out explicitly, as follows.

However, if the number of terms in the sum is large, summation notation saves a lot of time and space.

It should be clear from the graphs that adding up the areas of the
rectangles only approximates the area under the curve. However, by
increasing the number of subintervals the accuracy of the
approximation can be increased. All of the Maple commands described so
far in this lab permit a third argument to specify the number of
subintervals. The default is 4 subintervals. See the example below
for the area under *y*=*x* from *x*=0 to *x*=2 using the `rightsum`
command with 4, 10, 20 and 100 subintervals. (This area can be
calculated to be exactly 2.) Try it yourself with the `leftsum` and
`middlesum` commands.

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

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

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

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

In this example it appears that, as the number of subintervals increases, the rectangular approximation becomes more accurate. Of course, we realize that this is happening only because we knew from the start that the area of the triangular region being dealt with is 2.

Thu Mar 27 08:41:19 EST 1997