** Next:** Exercises
**Up:** Labs and Projects for
** Previous:** Labs and Projects for

Suppose *f*(*x*) is a non-negative, continuous function defined on some
interval [*a*,*b*]. Then by the area under the curve *y*=*f*(*x*) between
*x*=*a* and *x*=*b* we mean the area of the region bounded above by the
graph of *f*(*x*), below by the *x* axis, on the left by the vertical
line *x*=*a*, and on the right by the vertical line *x*=*b*.

All of the numerical methods in this lab depend on subdividing the
interval [*a*,*b*] into subintervals of uniform length. For example,
dividing the interval [0,4] into four uniform pieces produces the
subintervals [0,1], [1,2], [2,3], and [3,4].

**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. 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 the area under *y*=*x* from *x*=0 to *x*=2 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 two sections describe ways of assessing the accuracy of some rectangular approximations.

**upper sum approximation**- The height of the rectangle is the absolute maximum of
*f*(*x*) on the subinterval. **lower sum approximation**- The height of the rectangle is the absolute minimum of
*f*(*x*) on the subinterval.

It should be clear that, if the area being approximated has *A* square
units of area, then

lower sum *A* upper sum

In general, it is rather complicated to compute upper and lower sums.
However, if *f*(*x*) is monotonic, the situation is much easier. If
*f*(*x*) is increasing on the interval [a,b], then the upper sum is just
the right sum and the lower sum is just the left sum. In the last
example with *f*(*x*) = *x*, the right sums (which are upper sums) moved
down toward the value of *A* as the number of subintervals increased.
What happens with the left sums (which are lower sums) as n, the
number of subintervals, increases? The approximations of the area
using these two rules do not generate approximations which are
necessarily more or less accurate than the first three rules
presented. However, they are informative in that they give
lower and upper bounds on what the true area is so that, if the
function is not monotonic, at least a range for the true area
is available.

where *B* is the absolute maximum of |*f*''(*x*)| on [*a*,*b*]. In
practice, *B* is often approximated by a number *K* that is an upper bound
for *B*, that is *B* < *K*. For instance, if on
, *K* might be taken as 4. Do you see why? For more
complicated functions, Maple can be used to get a value for *K* that is
close to *B*.
Note that the error bound formula gives a worst case estimate, the
accuracy achieved for a given number of subintervals *n* may be much better than the guarantee
given by the formula.

12/10/1997