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Subsections
There are two main ways to think of the definite integral. The easiest
one to understand is as a means for
computing areas (and volumes). The second way the definite integral is
used is as a sum. That is, we use the definite integral to ``add
things up''. Here are some examples.
 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.
This lab is intended to introuduce you to Maple commands for computing
integrals, including applications of integrals.
The basic Maple command for computing definite and indefinite
integrals is the int command.
To compute the indefinite integral
with Maple:
> int(x^2,x);
Note that Maple does not include a constant of integration.
Suppose you wanted to compute the following definite
integral with Maple.
The command to use is:
> int(x^2,x=0..4);
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 solve or fsolve command. Once this is done, you can calculate the definite integral in Maple. An example below illustrates how this can be done in Maple by finding the area bounded by the graphs of
and :
> 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);
If a function is integrable over a closed interval , then the average value of , denoted ,
on this interval is:
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)/(51);
 Use Maple to compute the each of the following definite integrals:
 A)

(Note: To square a trig function put the
^
2 after the angle in parentheses.) Express your answer in analytic form.
 B)

(Note: The syntax for the function is .) Observe your answer in analytic form, then evaluate again and express your answer numerically.
 The region in the plane bounded by
and the axis over the interval
is a semicircle of radius 1. Approximate the area under the curve using a leftenpoint rule Riemann sum with 8 equally spaced rectangles and compare your answer to the area found by using a definite integral.
 Find the area of the region bounded by the curves
and
.
 Find the area of the region bounded by the curves
and
.
 Find the average velocity of a particle moving in one dimension with velocity
given in feet per second from to seconds.
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Dina J. SolitroRassias
20150916