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Much of Calculus II is devoted to the definite integral since that is the concept needed to deal with applications such as area, volume, work, etc. (as presented in Chapter 6 of the text). In order to know what definite integral to use in a given application, one needs to understand the Riemann sums that are used to define the definite integral (see Section 5.5 of the text) and to appreciate what each term in a Riemann sum stands for.

In this lab we consider Riemann sums associated with area problems
(see Sections 5.3 and 5.4 of the text for the appropriate background).
As you have seen, in an area problem, each term of the Riemann sum
represents the area of a rectangle and the Riemann sum itself gives an
approximation to the area under discussion. We now look in more
detail at these rectangular approximations. Note that all the
rectangular approximations considered in this lab depend on dividing
the interval in question into *n* subintervals of equal length.

**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 know 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 a pair of rectangular approximations.

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

3/25/1999