- 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 you 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);

It is also easy to use Maple to compute centroids and centers of mass. For example, suppose you wanted to compute the coordinates of the centroid of the region bounded above by the curve and below by the axis over the interval . The first thing to do is to plot the region.

> q := -3*x^2+3*x+36; > plot(q,x=-3..4);Once you have plotted the region, you can compute the area with the following command. Note the use of a label, since we'll use the area later.

> area := int(q,x=-3..4);Once we have the area, we can compute and as follows.

> x_bar := int(x*q,x=-3..4)/area; > y_bar := int(q^2,x=-3..4)/(2*area);A similar procedure can be used to find the centroid of a region in the plane that is bounded by the graphs of two functions. You just have to use the following formulas for and .

- Verify that the following equation is true, and then explain why
it is true, using properties of the integral.

- Find the area of the region bounded above by
and below by . Include a plot of the region.
- Find the centroid of the region from the previous exercise.
- In the last lab, you used the
`middlesum`rule to approximate the net distance traveled for an object whose velocity was given by . Use the Maple`int`command to get an exact answer for

Note that the answer comes out in terms of a function`FresnelS`. You can find out more about this function by asking your IA or executing the command> ?FresnelS

To obtain a numerical approximation to the integral, just put an`evalf`command on the outside of the`int`command.

2002-01-18