THE INTEGRATOR  is a free function integrator.  You need to read their instructions on entering input to use it effectively.  It is from Wolfram Research and powered by Mathematica.
	You can also compute integrals using Derive on your computer in class to check your answers or you can use the antiderivative feature on the Vanderbilt Toolkit.
	The Maths Online Gallery includes an intuitive approach to the definite integral.  After following this link, click on "Intuitively understanding the integral".
	Riemann Sum Applet
	Area Estimator
	IES Area Approximation by Rectangles
	For a nice set of review modules go to Visual Calculus: Antiderivatives, etc.
	Hotmath  You can look at free solutions to problems in exercise sets from a wide variety of mathematics textbooks including Calculus by Larson, Hostetler, Edwards, sixth edition.  Many of these exercises are identical to those in the seventh edition.  They now have chapters 1 - 6 and 85% of chapters 7 -12 completed for the seventh edition.  Only a few solutions are still free.
	Printable worksheets for graphical exercises can be found at mathgraphs.com.
	UC Davis has a site with a lot of problems and worked out solutions.
	Here is a Maple Worksheet with integration (and differentiation) examples.

4.1  Be able to evaluate indefinite integrals similar to those on page 255:15-42.  Be able to solve differential equations with initial conditions similar to those on page 256: 49-52, 55-62.  Be able to solve population growth and vertical motion problems similar to those on pages 256-57: 63-70.  Powerpoint Presentation on Antiderivatives

4.2  You will need to be able to find the upper and lower sums representing bounds on the area of a region bounded by the graphs of given equations and compute the limit of these sums to establish the area of the region.  An example of this would be Example 4 on pages 264-5 in your text or the following Visual Calculus demonstration.

 

In class I will use a left Riemann sum, right Riemann sum, and midpoint Riemann sum to approximate the area under the graph of y = x2 + 1 and above the x-axis with x between 0 and 2.  Here is a link to a Maple worksheet showing how I constructed the animations linked to in the previous sentence.  I also used a gif animator (to slow down the animation) and Quicktime Pro (to give some online control).  After approximating the area I will show how you can compute the exact area by computing a limit.  We will see that although the finite left Riemann sum would always be less than the area of the region and the finite right Riemann sum would always be greater than the area of the region, the limit for each will be the same.  You will need to be able to do this on your next exam.  The region whose area is being approximated and then computed exactly is pictured at the right in red.  Below are links to the animations looping forever without Quicktime.  More of the details on what I did in class. Left Riemann Sum  Right Riemann Sum  Midpoint Riemann Sum

 

4.3  You need to know the definition of Riemann Sum, definite integral, and when the definite integral represents the area of a region.  See The Area Problem and the Definite Integral.  You will need to know the Properties of Definite Integrals.  Here is another Riemann Sum applet. Here is yet one more very nice Riemann Sum applet and here is one that relates integration to finding the total distance traveled over an interval of time when the speed varies. (We will find out why the statement above is true in Section 4.4.) Click on the picture at the right to see an animation. Quicktime version of the animation

4.4  You need to understand the Fundamental Theorem of Calculus (and what theorem is used to prove it), the Mean Value Theorem for Integrals, the Definition of the Average Value of a Function on an Interval, and the Second Fundamental Theorem of Calculus.  Here is an applet relating integration to differentiation in the context of computing the total distance traveled over an interval of time by an object moving along a straight line with a varying speed.  You need to be able to apply the Fundamental Theorem of Calculus to evaluate definite integrals including evaluating definite integrals in computing the area of a region.  Check out this applet for a cumulative area function.  Here is a nice amplification on Buffon's Needle Experiment (P294: 100).

4.5  You will need to be able to demonstrate an appropriate substitution (u-substitution in class) in evaluating both indefinite integrals and definite integrals.  Be careful not to overlook something (involving the limits of integration) in using a substitution to evaluate a definite integral.  Note the change in the limits of integration in this applet for change of variables.  Here is another tutorial on the substitution technique, Karl's help in evaluating integrals using substitution, and drill problems involving the substitution technique (numbers 2, 4, 7, 9, and 10 relate to chapter 4).  At calc101.com/substitution worked out examples 1, 2, 3, and 5 would be good ones to look at.  Here is an applet you could use for practice integration problems with some instruction.

4.6  You will need to be able to articulate a geometric interpretation of how the Trapezoidal Rule and Simpson's Rule work.  You will need to be able to use your calculator or computer to perform numerical integration to approximate the value of a definite integral.  Here are two numerical integrators.  The first one includes a Trapezoidal Rule option and the second one includes both the Trapezoidal Rule and Simpson's Rule.  Here is another applet demonstrating Simpson's Rule and the Trapezoidal Rule.  Here is a wonderful Simpson's Rule applet by David Protas.

The pictures at the right show the "pieces of parabolas" used in applying Simpson"s Rule to the function f(x) = x3 - 2x2 - 3x + 10 over the interval [-2,2] with n = 2 (one parabola) and n = 4 (two parabolas).  In each of these cases Simpson's Rule would yield the exact value of the integral.

The picture at the right illustrates the Trapezoidal Rule with n = 4 (4 trapezoids).  The Trapezoidal Rule approximation to the integral given above with n = 4 would be 28 (compared to 29.333...).

Click here to see an animation depicting Simpson's Rule applied in approximating the integral given below.

Quicktime version

The exact answer is cos(8) + 8sin(8) -1.  In this example the graph of the function dips below the x-axis twice in the interval over which we are integrating.  What do you suppose the answer represents?

Click here to see an animation showing Simpson's Rule applied to a cubic function similar to the one in the pictures above.  Quicktime version  In this example the graph of the function also dips below the x-axis twice in the interval over which we are integrating.  What do you suppose the answer represents?

TAKE-HOME PROBLEMS FOR EXAM IV

Examples for Exam IV

Here is a preview of the area applications that will come in Calculus II including area between two curves.

Here is a Maple Worksheet with integration (and differentiation) examples.

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        Lane Vosbury, Mathematics, Seminole State College   email:  vosburyl@seminolestate.edu

        This page was last updated on 08/21/14          Copyright 2002          webstats