Suppose is a non-negative, continuous function defined on some interval . Then by the area under the curve between and we mean the area of the region bounded above by the graph of , below by the axis, on the left by the vertical line , and on the right by the vertical line .

All of the numerical methods in this lab depend on subdividing the interval into subintervals of uniform length. For example, dividing the interval into four uniform pieces produces the subintervals , , , and .

**left endpoint rule**- The height of the rectangle is the value of
the function at the left-hand endpoint of the subinterval.
**right endpoint rule**- The height of the rectangle is the value of
the function at the right-hand endpoint of the subinterval.
**midpoint rule**- The height of the rectangle is the value of the function 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. Take a moment to see
that the different rules choose rectangles which in each case will
either underestimate or overestimate the area.

> 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));

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 appoximations to the area under from to using
the `rightsum`
command with 4, 10, 20 and 100 subintervals. (As this region describes
a right triangle with height 2 and base 2, this area can be easily
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));

Since, in this trivial example, we knew that the area is exactly 2, it appears that, as the number of subintervals increases, the rectangular approximation becomes more accurate. What would happen with a not so trivial region? The next section describes a way of assessing the accuracy of the midpoint rule.

where is the absolute maximum of on , assuming that is continuous on that interval. In practice, is often approximated by a number that is an upper bound for , that is . For instance, if on , might be taken as 4. Do you see why? For more complicated functions, Maple can be used to get a value for that is close to or actually equal to . Note that the error bound formula gives a worst case estimate, the accuracy achieved for a given number of subintervals may be much better than the guarantee given by the formula.

2001-03-19