- Computing net or total distance traveled by a moving object.
- Computing average values, e.g. centroids and centers of mass, moments of inertia, and averages of probability distributions.

The command to use is shown below.

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

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));

To compute an indefinite integral with Maple, you just leave out the range for the limits of integration, as shown below.

> int(x^2,x);Note that Maple does not include a constant of integration.

You can also use the Maple `int` command with functions or
expressions you have defined in Maple.
For
example, suppose you wanted to find area under the curve of 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 as shown below.

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

You can also simply give the expression corresponding to 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 to choose one or the other, so you don't mix the syntax up.

> p := x*sin(x); > int(p,x=0..Pi);If you want to find the area bounded by the graph of two functions, you should first plot both functions on the same graph. You can then find the intersection points using either the

> f := x-> -x^2+4*x+6; > g := x-> x/3+2; > plot({f(x),g(x)},x=-2..6); > a := fsolve(f(x)=g(x),x=-2..0); > b := fsolve(f(x)=g(x),x=4..6); > int(f(x)-g(x),x=a..b);

Note that the average value is just a number. For example, suppose you wanted to compute the average value of the function over the interval . The following Maple command would do the job.

> int(-16*t^2+100*t,t=1..5)/(5-1);

- Use Maple to compute the following definite integrals.
- Find the area of the region bounded by the curves and by integrating with respect to . Find the area again by integration with respect to . Include a plot of the region.
- Suppose the velocities in miles per hour of two cars during a one hour race are

. If at the beginning of the race, both cars were at the mile mark,- Which car starts out traveling faster?
- Which car is traveling faster at the end of the race?
- Which car won the race? (Hint: Find their average velocities)

2002-11-06