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The Definite Integral


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. This lab is intended to introuduce you to Maple commands for computing integrals, including applications of integrals.

Definite and indefinite integrals with Maple

The basic Maple command for computing definite and indefinite integrals is the int command.

To compute the indefinite integral

\begin{displaymath}\int x^2   dx \end{displaymath}

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.

\begin{displaymath}\int_{0}^{4} x^2   dx \end{displaymath}

The command to use is:
> int(x^2,x=0..4);

Finding area

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 $f(x)=-x^2+4x+6$ and $g(x)=x/3+2$:
> 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);

Average value of a function

If a function $f$ is integrable over a closed interval $[a,b]$, then the average value of $f$, denoted $\bar{f}$, on this interval is:

\begin{displaymath}\bar{f} = \frac{1}{b-a} \int_{a}^{b} f(x)   dx \end{displaymath}

Note that the average value is just a number. For example, suppose you wanted to compute the average value of the function $s(t) =
-16t^2+100t$ over the interval $1 \leq t \leq 5$. The following Maple command would do the job.
> int(-16*t^2+100*t,t=1..5)/(5-1);


  1. Use Maple to compute the each of the following definite integrals:
    $\displaystyle \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{1}{\sin^2(x)+2} \, dx$ (Note: To square a trig function put the ^2 after the angle in parentheses.) Express your answer in analytic form.
    $\displaystyle \int_{-1}^{1} e^{-x^2-1}   dx$ (Note: The syntax for the $e^x$ function is $\exp(x)$.) Observe your answer in analytic form, then evaluate again and express your answer numerically.
  2. The region in the $xy$-plane bounded by $f(x)=\sqrt{1-x^2}$ and the $x$-axis over the interval $-1 \leq x \leq 1$ is a semi-circle of radius 1. Approximate the area under the curve using a left-enpoint rule Riemann sum with 8 equally spaced rectangles and compare your answer to the area found by using a definite integral.
  3. Find the area of the region bounded by the curves $f(x)=2\sqrt{x+10}$ and $\displaystyle g(x)=\frac{x^2}{4}-1$.
  4. Find the area of the region bounded by the curves $\displaystyle f(x)=\frac{1}{x^2+1}$ and $\displaystyle g(x)=\ln(x^2+1)$.
  5. Find the average velocity of a particle moving in one dimension with velocity $v(t)=98-3t+12\cos(2t)$ given in feet per second from $3$ to $7$ seconds.

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Next: About this document ... Up: lab_template Previous: lab_template
Dina J. Solitro-Rassias