In the example in the previous section, we saw that increasing the number of subintervals gave a better approximation to the area. In such a case, it seems reasonable that taking a limit as the number of subintervals goes to infinity should give the exact answer. This is exactly the idea in defining the definite integral. Of course, the actual definition of the definite integral involves more general sums than the ones we have been talking about, but the idea is the same.

When this limit as the number of subintervals goes to infinity exists, we have special notation for this limit and write it as

We will learn later on that if the function *f*(*x*) is continuous on
the interval [*a*,*b*], then this limit always exists and we can write

where *A* is the area under the curve *y*=*f*(*x*) between *x*=*a* and *x*=*b*.

The Fundamental Theorem of Calculus (FTOC) provides a connection
between the definite integral and the indefinite integral we studied
earlier. That is, the FTOC provides a way to evaluate definite
integrals if an anti-derivative can be found for *f*(*x*). We'll spend a
lot of time later in the course developing ways to do this, but for
now we'll let Maple do the work and won't worry too much about how it
is done.

The basic maple command for performing definite and indefinite
integrals is the `int` command. The syntax is very similar to
that of the `leftsum` and `rightsum` commands, except
you don't need to specify the number of subintervals. This should make
sense, if you recall that the definite integral is defined as a limit
of a rectangular sum as the number of subintervals goes to
infinity. In the section on rectangular approximations, we used two
examples. The first was the function on the interval [0,4]
and the second was the function *y*=*x* on the interval [0,2]. We
would express the areas under these two curves with our integral
notation as

and

Using Maple, we would compute these two definite integrals as shown below.

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

> int(x,x=0..2);

Notice that Maple gives an exact answer, as a fraction. If you want a
decimal approximation to an integral, you just put an `evalf`
command around the `int` command, as shown below.

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

In the exercises, we'll ask you to compare rectangular approximations
to integrals. In doing so, you'll need to apply several commands to
the same function. To save typing and prevent errors, you can define the
function as a function or an expression in Maple first and then use it
in subsequent `int`, `leftsum`, etc. commands. For
example, suppose you were given the function on the
interval . Then you can define this function in Maple with
the command

> f := x -> x*sin(x);

and then use this definition to save typing as shown below.

> int(f(x),x=0..Pi);

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

You can also simply give the expression corresponding to *f*(*x*) a
label in Maple, and then use that label in subsequent commands as
shown below. However, notice the difference between the two
methods. You are urged you to choose one or the other, so you don't
mix the syntax up.

> p := x*sin(x);

> int(p,x=0..Pi);

Fri Jan 17 13:12:57 EST 1997